Analysis

Table of Contents

Defintions

General

The support of a real-valued function analysis_39852e5ec4f953f23fa3efc2c6557ea8a7e7114f.png is given by

analysis_f028b704e0e2d5d43142077aa7737e30c9b1f79f.png

Let analysis_8c8d4992aa385c22d79f66db8edced75a661b728.png be a non-empty set, sometimes reffered to as the index set.

A symmetric function analysis_a6ae22da5537f0954e45219149ef22468e94925b.png is called a positive-definite kernel of analysis_8c8d4992aa385c22d79f66db8edced75a661b728.png if

analysis_f7581fdb48b85a53c6a6b770ee6d91c60f038eea.png

holds for any analysis_84feeb83da8919c3f995550cdfb7e928ee8b390d.png, analysis_195236150174d489bcc4f7f60599c40da8f4f50c.png, analysis_27633ac8c8aa0ae9d092b03465d6922c89a928d0.png.

Or more generally, for some field real or complex field analysis_f947cfd1ce386c5e05e97c005ab84e6b63f9616a.png, a function analysis_ab2162a8e45875cfcd3ff16e9c0b169de768c4bc.png is called a kernel on analysis_8c8d4992aa385c22d79f66db8edced75a661b728.png if there exists a analysis_e999b47e39f80ee0a17ed49d7455db2a08812ab3.png and a map analysis_44da885fdad5f1a471bcb221724466a4c1f2f40c.png such that for all analysis_54b7ae2b58b08839e638b82168a2ee7f3c30cc09.png we have

analysis_15c0c0641c9fa38bf6cb51f0d6b4e319a3fe8787.png

In machine learning context, you'll often see analysis_05594e8ce8f52728892c4c673d8bc28149b4b24d.png be called a feature map and analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png a feature space of analysis_1641d18cc980f8db14cdff95d7417a8526eef446.png.

Note the swapping of order in the kernel and inner product above; this is only necessary in the case of a complex Hilbert space, where the inner product is sesquilinear:

analysis_6be81aca65b889a3e5536fe490e6c2c16656540a.png

Lp space

analysis_5bda34178cc82d1588df2a197af81c5f47d96d87.png spaces are function spaces defined using a generalization of the p-norm for a finite-dimensional vector space.

p-norm

Let analysis_4663bc0a6c334a543bdb9be2341fb5d07e33bded.png be a real number. Then p-norm (also called the analysis_2bd272d1d5879c47adcb2a838c01287593bf0c7e.png -norm) of vectors analysis_bb43008f573e07f365326e6bc76023e2e81628a8.png, i.e. over a finite-dimensional vector space, is

analysis_48bb26b9dee2292c78e8c7737384c2f891a12d85.png

Banach space

A Banach space is a vector space with a metric that:

  • Allows computation of vector length and distance between vectors (due to the metric imposed)
  • Is complete in the sense that a Cauchy sequence of vectors always converge to a well-defined limit that is within the space

Sequences

Sequences of real numbers

Definition of a convergent sequence

analysis_3d0b17eab804b54fa05510288ce7f5ae0f9de588.png

analysis_c6a2b1467c0e7a00a40daf353d47b96c6292ebdb.png such that analysis_d76400b96cefe9bbdc4141db3825780299e7d812.png analysis_0511a007359502efedd24d487bc7b8b378ea17f6.png such that analysis_776d996567a04d1d1c3f292ec0da017f8dc1910a.png

Bounded sequences

A bounded sequence analysis_0716d32982ef5d050c13ead8b974fdc507dc7496.png such that

analysis_fb88b371492f94927ade2fe6e18b45d9dec894da.png

Cauchy Sequence

A sequence of points analysis_ec30a2df912f2cd1cca69a64174945bbe801a45a.png is said to be Cauchy (in analysis_492d525117d0dcc93d066c8759f46b98cf9980ca.png) if and only if for every analysis_04c85e01ffbb8ac64d83aef1660930fa71b72d75.png there is an analysis_cd749db889b6ca80f8eed8b7f72453297d916c84.png such that

analysis_0bd67bb93eaa4b98574b159ce38937966f409827.png

For a real sequence this is equivalent of convergence.

TODO Uniform convergence

Series of functions

Pointwise convergence

Let analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png be a nonempty subset of analysis_492d525117d0dcc93d066c8759f46b98cf9980ca.png. A sequence of functions analysis_e5acff82b29ef83779f0b8802f221581af6629c7.png is said to converge pointwise on analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png if and only if analysis_50ac5fbb4696ab0322d8bc460371a8a96fb92e7b.png exists for each analysis_a75b7783a1f07c2000e3a9ef2d944140b38f6435.png.

We use the following notation to express pointwise convergence :

analysis_d206f8bbdf2ecadd2e77fe2533dd26fcd427e19a.png

Remarks
  • The pointwise limit of continuous (respectively, differentiable) functions is not necessarily continuous (respectively, differentiable).
  • The pointwise limit of integrable functions is not necessarily integrable.
  • There exist differentiable functions analysis_617b87515c58d75e4faeff2a8dbc4b795e512b7c.png and analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png such that analysis_86fff707df410a65631e9a7f612d32ddf7486c48.png pointwise on $[0, 1], but

analysis_90e0a3e0e3c9040734c90e605363bbda04a047c7.png

  • There exist continuous functions analysis_617b87515c58d75e4faeff2a8dbc4b795e512b7c.png and analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png such that analysis_86fff707df410a65631e9a7f612d32ddf7486c48.png pointwise on $[0, 1] but

analysis_d38bcabe30feffa9811735d237c5bb68f286f3d4.png

Uniform convergence

Let analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png be a nonempty subset of analysis_492d525117d0dcc93d066c8759f46b98cf9980ca.png. A sequence of functions analysis_e5acff82b29ef83779f0b8802f221581af6629c7.png is said to converge uniformly on analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png to a function analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png if and only if for every analysis_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png.

Continuity

Lipschitz continuity

Given two metric spaces analysis_f5f5b2f282fe2f88cc63368d78ebb80eb1ac38c3.png and analysis_989aa42942b6f7825ee1ee82b6ea876c5a2c365c.png, where analysis_d78f82f936b1e2e93688cff4bb70a7516696bda6.png denotes the metric on analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png, and same for analysis_93ad5a317bb284b4a59ac6c2c3422744bfcd1f51.png and analysis_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png, a function analysis_824489486a0fc6b011b40e15f1e2e96f7e410566.png is called Lipschitz continuous if there exists a real constant analysis_8204182114ab094a99dbd3a786c5b31390379032.png such that

analysis_3bd8bf4cf28b9bc94cbce8bc0c79c0363cfb0d72.png

Where the constant analysis_1641d18cc980f8db14cdff95d7417a8526eef446.png is referred to as the Lipschitz constant for the function analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png.

If the Lipschitz constant equals one, analysis_905f404fab412feec18ca89d266ef27504783d6a.png, we say analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is a short-map.

If the Lipschitz constant analysis_1641d18cc980f8db14cdff95d7417a8526eef446.png is

analysis_1a19ba9de9ed37ef36be3bc1ab9542c7cd263b01.png

we call the map analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png a contraction or contraction mapping.

A fixed point (or invariant point ) analysis_fe3d8b42293a7c7119494c5a50c513c0191b2d63.png of a function analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is an element of the function's domain which is mapped onto itself, i.e.

analysis_2456c3f747e94536b3b37104ae785b746d8feb3a.png

Observe that if a function analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png crosses the line analysis_e06b484ee2b1375e0428320e18703e830425f7db.png, then it does indeed have a fixed point.

The mean-value theorem can be incredibly useful for checking if a mapping is a contraction mapping, since it states that

analysis_363cca3e2b1c144053253dbe141b7d3621c1877d.png

for some analysis_62a73ee5a8de6d9edb18f8f2c6d60795777228a8.png.

Therefore, if there exists analysis_cc6b5c35669c3b0901ef37cf78827ca69227d7b2.png such that analysis_3187cbe52040a6bcd95516ef884c261433dbcd98.png for all analysis_d16f4fe600dfe23f9c4c2594f9c1043341f6ea1a.png then we clearly know that

analysis_c0707d2a11d886b1d46ac8e5e4074e6ffb832fb4.png

hence it's a contraction.

Hölder continuity

A real- or complex-valued function analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png on a d-dimensional Euclidean space is said to satisfy a Hölder condition or is Hölder continuous if there exists non-negative analysis_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png and analysis_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png such that

analysis_87c3fe7fbe63efd4bed067034c251ab126f2226a.png

for all analysis_ce64de6a9b96397cd349dd5414fd9a75519e368a.png in the domain of analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png.

This definition can easily be generalized to mappings between two different metric spaces.

Observe that if analysis_748079db104c9df39eddf739667af5bf6cd790d8.png, we have Lipschitz continuity.

Càdlàg function

Let analysis_bdb741bb1f4abc6a802521e096911386d8c3f7af.png be a metric space, and let analysis_f9bf6d664d9d2a949a6d9944de07b272a08dcb57.png.

A function analysis_7cd06de55c6fd32dc68c72a94b780ce10291c661.png is called a càdlàg function if, for very analysis_e02303492e98c08e8abcf40b0e26f9deb496216d.png:

  1. The left limit

    analysis_d9e190378ad928c915cf1a022e655512ef314162.png

    exists.

  2. The right limit

    analysis_a23004f1c6bac09e7e5f2148b5c85d3877cae54b.png

    exists and equals analysis_9a6c9e4e5580d33233d25060fb4b1ca77fc8b57a.png

That is, analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is right-continuous with left limits.

Affine space

An affine space generalizes the properties of Euclidean spaces in such a way that these are independent of concepts of distance and measure of angles, keeping only properties related to parallelism and ratio of lengths for parallel line segments.

In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. We instead work with displacement vectors, also called translation vectors or simply translations, between two points of the space.

More formally, it's a set analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png to which is associated a vector space analysis_5621442eb84314df5b6d4f90193ccefd86434bb0.png and a transistive and free action analysis_07aa11d29578308bc0642bd6f5d3e959f8d85582.png of the additive group of analysis_5621442eb84314df5b6d4f90193ccefd86434bb0.png. Explicitly, the definition above means that there is a map, generally denoted as an addition

analysis_a7a388804b391373fb6f42b8cbbd6a62c8256810.png

which has the following properties:

  1. Right identity: $∀ a ∈ A, a + 0 = 0 $
  2. Associativity:

    analysis_38d2be25729e3c81c065df70984832f2bdcd6657.png

  3. Free and transistive action: for every analysis_059a13dab69c3d1523accbe2cb781457a950d07b.png, the restriction of the group action to analysis_70d425437d44f36a7dd9c28b54479061a4043b33.png, the induced mapping analysis_4424d7d9cfc3f8812ff1f8c1adcfd6e1c462928b.png is a bijection.
  4. Existence of one-to-one translations: For all analysis_1bf7e0a84bccb025565bef19536afe2a5a2a681f.png, the restriction of the group action to analysis_7acca89ca1cf53e77f15a7030253ef58987c0cd3.png, the induced mapping analysis_711bf97782bc8861c9985f6f35de9e22c47a8270.png is a bijection.

This very rigorous definition might seem very confusing, especially I remember finding

analysis_8e27b5dfe02921a7af855213d4ec6577db8b6b05.png

to be quite confusing.

"How can you map from some set analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png to itself when clearly the LHS contains an element from the vector space analysis_5621442eb84314df5b6d4f90193ccefd86434bb0.png!?"

I think it all becomes apparent by considering the following example: analysis_bd57881e4ed69c0133b3fca5ee7815e45cee3b8e.png and analysis_da7b2b292c200a18457ec3b985a3196fbb1f14ce.png. Then, the map

analysis_ada10d7fdc82b8841f9ea80c1a99b8c61e79ec4b.png

Simply means that we're using the structure of the vector space analysis_6abaf46f98f7e7ed4cad77ce3c02ca6595390335.png to map an element from analysis_004097ff73cb85a0f596c8a3b60218ece0e16be1.png to analysis_004097ff73cb85a0f596c8a3b60218ece0e16be1.png!

Could have written

analysis_2562d10ab58b6a8c33bf71a02617bfe5d9023839.png

to make it a bit more apparent (but of course, a set does not have any inherit struct, e.g. addition).

Schwartz space

The Schwartz space analysis_edc8c26154e55922c8a59cf648600dd1f1f1beda.png is the space of all analysis_142e99eaf45f1872ae84813baea3d90f924b5333.png functions analysis_541a8541573e130dc310c688d0524f6e8fc0a5e3.png on analysis_004097ff73cb85a0f596c8a3b60218ece0e16be1.png s.t.

analysis_551365913e8b2f53b864ca80bbb1b1c736fa9811.png

for all analysis_dcbfd7fd00b73ed5f0c8296aa603f19bf57c9e87.png.

Here if analysis_214c5c63c09f52847a73c3ea8f29855d186564f5.png then analysis_fb99d05f3e79ebbd2216cc77cadc447080a30b43.png and

analysis_f1b1cf50ba919d3ca48bd2cf4bf644ddf1c3746f.png

An element of the Schwartz space is called a Schwartz function.

Theorems

Cauchy's Theorem

This theorem gives us another way of telling if a sequence of real numbers is Cauchy.

Let analysis_3e0d749f8b54de075cf4b60155d416c2c52e85fe.png be a sequence of real numbers. Then analysis_3e0d749f8b54de075cf4b60155d416c2c52e85fe.png is Cauchy if and only if analysis_3e0d749f8b54de075cf4b60155d416c2c52e85fe.png converges (to some point analysis_3778a663528a3e4db5ce811ed0c6cd9f5831579f.png in analysis_b8615deb99f6b822a41b83e0702f97500dd423f2.png).

Suppose that analysis_3e0d749f8b54de075cf4b60155d416c2c52e85fe.png is Cauchy. Given analysis_a0a4c4cfaefb07a6304f5c219ec22f561ac325ab.png, choose analysis_cd749db889b6ca80f8eed8b7f72453297d916c84.png such that

analysis_2d463463111617f4301c4b9e5ef89db75dd6ab5a.png

By the Triangle Inequality,

analysis_6474f09c11f2d6e7fcc19cced1285a44f91344ee.png

Therefore, analysis_3e0d749f8b54de075cf4b60155d416c2c52e85fe.png is bounded by analysis_217f00dbe024300023e8cbdae739686e4eddc063.png.

By the Bolzano-Weierstrass Theorem

Telescoping series

analysis_01854ac03c89b290614463f5a514ee77e8e5e938.png

Bolzano-Weierstrass Theorem

A sequence analysis_2fd852297613d1dbb919116a49e00ed673f96c16.png of sets is said to be nested if

analysis_d2d06f50f8fa5fc7996dd6d8e8665b789c79f15b.png

If analysis_2fd852297613d1dbb919116a49e00ed673f96c16.png is a nested subsequence of nonempty bounded intervals, then

analysis_d086003975f34b8b0507b5d3af028bf5fdfff6ee.png

is non-empty (i.e. contains at least one number).

Moreover, if analysis_e3d2f1061d7daa7d52ea2d3f7ad5ef5ebd74b039.png then analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png contains exactly one number (by non-emptiness of analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png).

Each bounded sequence in analysis_004097ff73cb85a0f596c8a3b60218ece0e16be1.png has a convergent subsequence.

Assume that analysis_3778a663528a3e4db5ce811ed0c6cd9f5831579f.png is the lower and analysis_f8ba0bcc022c6d477cbfde325f315454468205bb.png the upper bound of the given sequence. Let analysis_35c78a8ef4f641c59e54e02bab73db9e28f367be.png.

Divide analysis_8a909f6c841116d84ca8b91013fc256f0dfb7a92.png into two halves, analysis_1a84f7224feced5922007ffa4b165bc2e6d4d942.png and analysis_d7a09c4d38f41b9993318b04d632c673897f843d.png:

analysis_ba029dad3dccc5841fad5cfb92ddd33723f8f41e.png

Since analysis_2013998be62c9aca5d4f0459304a4a17fc51f8e1.png at least one of these intervals contain analysis_3b985da1e55f724a258884652d2772d6f66f75c5.png for infinitively many values of analysis_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png. We denote the interval with this property analysis_54a5e25a2aeee01dbfc339faa7742a04aee3770d.png. Let analysis_dd2d38a3e214f85182f4d19aaf6458c171a42fd5.png be such that analysis_f1977e3d7e1d8769b8c542e526ca77bbbf926da0.png.

We proceed by induction. Divide the interval analysis_421a3edb098d5ba9c9936080c05660d868bfce20.png into two halves (like we did with analysis_8a909f6c841116d84ca8b91013fc256f0dfb7a92.png). At least one of the two halves will contain infinitively many analysis_3b985da1e55f724a258884652d2772d6f66f75c5.png, which we denote analysis_c3ca649aa58416b9f6ecbc742f7b448ba023854d.png. We choose analysis_4d886782e948c7c439dd23977734ed9db0406462.png such that analysis_701da9c7081dc0647c0b06888c126a5ec1653229.png.

Observe that analysis_19891af493aa68030b37de7c0b4341f7b60f333e.png is a nested subsequence of boudned and closed intervals, hence there exists analysis_b0706adb445a2c20a4d236cf389c6138cf2c5ccb.png that belongs to every interval analysis_1c9ba208f7b3c6266197ccf6ed139f510737c985.png.

analysis_31db975721be26f5d6f8f7ea7daf5fce70635548.png

By the Squeeze Theorem analysis_2f7db2201a10875a5d160322920454006b368bfa.png as analysis_2703c2b1ae53a3e5be064c15b4bb53f564ae9ac4.png.

Triangle Inequality

analysis_ace8284bbcf6b79fe5231d13f1a589e0b877e792.png

Mean Value Theorem

If a function analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is continuous on the closed interval analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png , and differentiable on the open interval analysis_4dbac29c08ad38dc09a53b1d71656a5e05b88888.png , then there exists a point analysis_cc6b5c35669c3b0901ef37cf78827ca69227d7b2.png in analysis_4dbac29c08ad38dc09a53b1d71656a5e05b88888.png such that:

analysis_cac7d10ab3ad0855f7729c8577d84208acfefa16.png

Rolle's Theorem

Suppose analysis_ace2bca881e8ad00894c8bc9404c9d32b19d7e07.png with analysis_9c2f16862ccf2498b205c4cb227c1866a266f2c3.png. If analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is continuous on analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png, differentiable on analysis_4dbac29c08ad38dc09a53b1d71656a5e05b88888.png and analysis_905a70c8d35fb5fb222ea2110243173819673b9d.png then analysis_291a912739bf013a4e90f6bad01e67ffae49fcd8.png for some analysis_4a8d2b41b961b0f62afb3e6fd15b588909b2538f.png.

Intermediate Value Theorem

Consider an interval analysis_7fffadbade889774e19af9f1f8c0b7927e16f3d7.png on analysis_492d525117d0dcc93d066c8759f46b98cf9980ca.png and a continuous function analysis_b71b9ff6ed85a48c1d2ea44846e83f4d4d43d1d9.png. If analysis_8b0e7a5ea4d56f5afdac3c2658ca87bd50b0ad6f.png is a number between analysis_f789c9a985bfda0b04e6e6e362810952c80a3ba3.png and analysis_2406968d35241026660ef1db7c5a40b7fb232930.png, then

analysis_a7ab09a1d99e366fe1e5f4bb9c42bbe3ddff5ba3.png

Useful identities

Upper bound on abs of sin(x)

analysis_d69bdffbabf866e14f2e13f6155daf5356557db3.png

due to the Mean Value Theorem.

M-test

Let analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png be a nonempty subset of analysis_492d525117d0dcc93d066c8759f46b98cf9980ca.png and

analysis_f1c6c7f571917e5062bd97c03167517990967334.png

and suppose

analysis_b3dddd6364ce73469d8036eb91bf8e01f55349ce.png

(i.e. series is bounded ). If analysis_a5d64815fd498607b53b304f24e894573632129c.png for analysis_a75b7783a1f07c2000e3a9ef2d944140b38f6435.png, then

analysis_dc1c32aab5e3e90c80f01866aa0bbf400bdb271f.png

converges absolutely and uniformly on analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png.

Fixed Point Theory

Banach Fixed Point Theorem

Let analysis_f7eec20146ea61ba7ca25392fa7ea5297cb502ce.png be a be non-empty complete metric space with a contraction mapping analysis_287bf5beddadd0da8587c5ba38d70f9fb077ae74.png. Then analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png admits a unique fixed-point analysis_fe3d8b42293a7c7119494c5a50c513c0191b2d63.png in analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

Furthermore, analysis_fe3d8b42293a7c7119494c5a50c513c0191b2d63.png can be found as follows:

Start with an arbitrary element analysis_c462c980a45481116745a8647094b2b1d245df0f.png in analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and define a sequence analysis_3e0d749f8b54de075cf4b60155d416c2c52e85fe.png by analysis_bdd120d7d4cd75d81a3be7a340acd70f6f9d111a.png, then analysis_3cfada7b8983d9db96b7334a833e0af91e526014.png

When using this theorem in practice, apparently the most difficult part is to define the domain analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png such that analysis_99a1bba1a9d7f358f20a1891df6deee306b57967.png.

Fundamental Contraction Inequality

By the triangle inequality we have

analysis_830ab28622c9e38418574ba0e525819e27d0a220.png

Where we're just using the fact that for any two different analysis_7a84c9a383f9772338016d101ccc096be06af784.png and analysis_eb83d466c7d035356e9f39998f357cee73da1e26.png, analysis_8a62b803144e38fe1586e88f8e706eb845651424.png is at least less than analysis_5a95c7ca0110f403f48b516cf76e38c829ea6a93.png by assumption of analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png being a contraction mapping.

Solving for analysis_9a1fa803b4f6996da4aa10b9eca1ce79b39f14a0.png we get

analysis_35bf624215af81757a72c94a42807ac3610efe2f.png

Measure

Definition

A measure on a set is a systematic way of defining a number to each subset of that set, intuitively interpreted as size.

In this sense, a measure is a generalization of the concepts of length, area, volume, etc.

Motivation

The motivation behind defining such a thing is related to the Banach-Tarski paradox, which says that it is possible to decompose the 3-dimensional solid unit ball into finitely many pieces and, using only rotations and translations, reassemble the pieces into two solid balls each with the same volume as the original. The pieces in the decomposition, constructed using the axiom of choice, are non-measurable sets.

Informally, the axiom of choice, says that given a collecions of bins, each containing at least one object, it's possible to make a selection of exactly one object from each bin.

Measure space

If analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is a set with the sigma-algebra analysis_017948b866be67b1a8e56a6b5f8848f823410c34.png and the measure analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png, then we have a measure space .

Sigma-algebra

Let analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be some set, and let analysis_6f356776de41b193d9421916df1afc027221c49d.png be its power set. Then the subset analysis_560131fab33600e7156a7fdb2a816b224a46e25c.png is a called a σ-algebra on analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png if it satisfies the following three properties:

  1. analysis_80a4e18121ac309a52c5db8c79e7b082a5d1819d.png
  2. analysis_017948b866be67b1a8e56a6b5f8848f823410c34.png is closed under complement: if analysis_d74935f39042b03a6ff552bcd4802908fffe3d5d.png
  3. analysis_017948b866be67b1a8e56a6b5f8848f823410c34.png is closed under countable unions: if analysis_c1dc820d66c3a8e96360ca48754b7b9a39628344.png

These properties also imply the following:

  • analysis_3416854bb7897915e8900878d7fdd4ce77fab8f1.png
  • analysis_017948b866be67b1a8e56a6b5f8848f823410c34.png is closed under countable intersections: if analysis_8b7e76de03d521b4dd4cbba867549ef707be4cd1.png

A measure analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png on a measure space analysis_8c9410961da365661c5bac7225665630a902ea74.png is said to be sigma-finite if analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png can be written as a countable union of measurable sets of finite measure.

Borel sigma-algebra

Any set in a topological space that can be formed from the open sets through the operations of:

  • countable union
  • countable intersection
  • complement

is called a Borel set.

Thus, for some topological space analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png, the collection of all Borel sets on analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png forms a σ-algebra, called the Borel algebra or Borel σ-algebra .

Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called a Borel measure.

Lebesgue sigma-algebra

Basically the same as the Borel sigma-algebra but the Lebesgue sigma-algebra forms a complete measure.

  • Note to self

    Suppose we have a Lebesgue mesaure on the real line, with measure space analysis_02b3b2e9358326ceefbd0f9e5ac4014d40fba374.png.

    Suppose that analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is non-measurable subset of the real line, such as the Vitali set. Then the analysis_4a04566b671f768402c9b91f57707ea1523d782b.png measure of analysis_9574d40571bf38a0136479fd0d504d2e719c41f2.png is not defined, but

    analysis_eed109b95a9038895a22ec05946d227a4bb22b20.png

    and this larger set ( analysis_3ff49ed7ddd4f67d576c9ce3905bdbe1bfee4047.png ) does have analysis_4a04566b671f768402c9b91f57707ea1523d782b.png measure zero, i.e. it's not complete !

  • Motivation

    Suppose we have constructed Lebesgue measure on the real line: denote this measure space by analysis_02b3b2e9358326ceefbd0f9e5ac4014d40fba374.png. We now wish to construct some two-dimensional Lebesgue measure analysis_4a04566b671f768402c9b91f57707ea1523d782b.png on the plane analysis_e4f375c26796781f71b7ae3026445db617a6e78b.png as a product measure.

    Naïvely, we could take the sigma-algebra on analysis_e4f375c26796781f71b7ae3026445db617a6e78b.png to be analysis_f511b32cdc5972418b6443b060202d9dc94e42c2.png, the smallest sigma-algebra containing all measureable "rectangles" analysis_5a83e273444069ea4d94185dfcb1f38b0e9b2413.png for analysis_34fd1ec18df7dc8e4893bc7b138da6231940004d.png.

    While this approach does define a measure space, it has a flaw: since every singleton set has one-dimensional Lebesgue measure zero,

    analysis_2fd48313f6009dfc982b60c4792f3d2667872916.png

    for any subset of analysis_492d525117d0dcc93d066c8759f46b98cf9980ca.png.

    What follows is the important part!

    However, suppose that analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is non-measureable subset of the real line, such as the Vitali set. Then the analysis_4a04566b671f768402c9b91f57707ea1523d782b.png measure of analysis_9574d40571bf38a0136479fd0d504d2e719c41f2.png is not defined (since we just supposed that analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is non-measurable), but

    analysis_eed109b95a9038895a22ec05946d227a4bb22b20.png

    and this larger set ( analysis_3ff49ed7ddd4f67d576c9ce3905bdbe1bfee4047.png ) does have analysis_4a04566b671f768402c9b91f57707ea1523d782b.png measure zero, i.e. it's not complete !

  • Construction

    Given a (possible incomplete) measure space analysis_85836c3c18ee764fbdef801c3631f37f2456dcf7.png, there is an extension analysis_af9d9087db2dc793391a11589265145931b00dc3.png of this measure space that is complete .

    The smallest such extension (i.e. the smallest sigma-algebra analysis_f992e42c2c120dd425fbadf62f2573acb2187637.png ) is called the completion of the measure space.

    It can be constructed as follows:

    • Let analysis_ca2bcc80309e331f71dd2c1e1f4788cbc048c242.png be the set of all analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png measure zero subsets of analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png (intuitively, those elements of analysis_ca2bcc80309e331f71dd2c1e1f4788cbc048c242.png that are not already in analysis_017948b866be67b1a8e56a6b5f8848f823410c34.png are the ones preventing completeness from holding true)
    • Let analysis_f992e42c2c120dd425fbadf62f2573acb2187637.png be the sigma-algebra generated by analysis_017948b866be67b1a8e56a6b5f8848f823410c34.png and analysis_ca2bcc80309e331f71dd2c1e1f4788cbc048c242.png (i.e. the smallest sigma-algreba that contains every element of analysis_017948b866be67b1a8e56a6b5f8848f823410c34.png and of analysis_ca2bcc80309e331f71dd2c1e1f4788cbc048c242.png)
    • analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png has an extension to analysis_f992e42c2c120dd425fbadf62f2573acb2187637.png (which is unique if analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png is sigma-finite), called the outer measure of analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png, given by the infimum

    analysis_d39cc89a7d78a75ce7924aec030fcaf2e2bf7c4e.png

    Then analysis_af9d9087db2dc793391a11589265145931b00dc3.png is a complete measure space, and is the completion of analysis_85836c3c18ee764fbdef801c3631f37f2456dcf7.png.

    What we're saying here is:

    • For the "multi-dimensional" case we need to take into account the zero-elements in the resulting sigma-algebra due the product between the 1D zero-element and some element NOT in our original sigma-algebra
    • The above point means that we do NOT necessarily get completeness, despite the sigma-algebras defined on the sets individually prior to taking the Cartesian product being complete
    • To "fix" this, we construct a outer measure analysis_dc3dda5fbf9d111629db86b3e3690b394b7d7d32.png on the sigma-algebra where we have included all those zero-elements which are "missed" by the naïve approach, analysis_f992e42c2c120dd425fbadf62f2573acb2187637.png

Product measure

Given two measurable spaces and measures on them, one can obtain a product measurable space and a product measure on that space.

A product measure analysis_201435ab560b9faa34976b4b74b3c3e80c29a5ae.png is defined to be a measure on the measurable space analysis_4abd990c6ed8a4f055141a7f7675305ce09d585a.png, where we've let analysis_60b59bddc7486649ac40ab7d4fbb8bf58c4db3eb.png be the algebra on the Cartesian product analysis_426801f6f272b4f2bda3fd2f4bb41a4937108eb0.png. This sigma-algebra is called the tensor-product sigma-algebra on the product space.

A product measure analysis_201435ab560b9faa34976b4b74b3c3e80c29a5ae.png is defined to be a measure on the measurable space analysis_4abd990c6ed8a4f055141a7f7675305ce09d585a.png satisfying the property

analysis_8d0228c1c42ef424b8d3ddb8720621a9d3f8674e.png

Complete measure

A complete measure (or, more precisely, a complete measure space ) is a measure space in which every subset of every null set is measurable (having measure zero).

More formally, analysis_85836c3c18ee764fbdef801c3631f37f2456dcf7.png is complete if and only if

analysis_840395bef2579ed354e2046d29c7bd70e2b0cfb2.png

Lebesgue measure

Given a subset analysis_f9bf6d664d9d2a949a6d9944de07b272a08dcb57.png, with the length of a closed interval analysis_8de9f615a64b2e89e1f5c33cd1153517015b6a73.png given by analysis_f0ae4401c60408cdcc680328c83e65725e291a1f.png, the Lebesgue outer measure analysis_9c5a838dfd45201e3f983614c4fbae1348940f16.png is defined as

analysis_58f487c0339fe1bed203ebbc778c004e4b1dbf07.png

The Lebesgue measure is then defined on the Lebesgue sigma-algebra, which is the collection of all the sets analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png which satisfy the condition that, for every analysis_8ce3a5b1041a76dd221dc188f1eac461178977ab.png

analysis_ecdfc5ea1fc4dc4ad26cf520ea7c633eaa0b90cd.png

For any set in the Lebesgue sigma-algrebra, its Lebesgue measure is given by its Lebesgue outer measure analysis_0667014a76698d6d5aea1a8fe8a96a0f87e62acd.png.

IMPORTANT!!! This is not necessarily related to the Lebesgue integral! It CAN be be, but the integral is more general than JUST over some Lesgue measure.

Intuition
  • First part of definition states that the subset analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png is reduced to its outer measure by coverage by sets of closed intervals
  • Each set of intervals analysis_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png covers analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png in the sense that when the intervals are combined together by union, they contain analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png
  • Total length of any covering interval set can easily overestimate the measure of analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png, because analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png is a subset of the union of the intervals, and so the intervals include points which are not in analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png

Lebesgue outer measure emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets. Intuitively, it is the total length of those interval sets which fit analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png most tightly and do not overlap.

In my own words: Lebesgue outer measure is smallest sum of the lengths of subintervals analysis_1c9ba208f7b3c6266197ccf6ed139f510737c985.png s.t. the union of these subintervals analysis_1c9ba208f7b3c6266197ccf6ed139f510737c985.png completely "covers" (i.e. are equivalent to) analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png.

If you take an a real interval analysis_7fffadbade889774e19af9f1f8c0b7927e16f3d7.png, then the Lebesge outer measure is simply analysis_718d86e72a0189b0ec0bf9d114c6f5b030fdf1a0.png.

Lebesgue Integral

The Lebesgue integral of a function analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png over a measure space analysis_85836c3c18ee764fbdef801c3631f37f2456dcf7.png is written

analysis_dd35ce19f3448e5b556b5e054674644f232e5ce4.png

which means we're taking the integral wrt. the measure analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png.

riemann_vs_lebesgue_integral.png

Special case: non-negative real-valued function

Suppose that analysis_95582f089b0a75752fa0a50d824485f0482bd980.png is a non-negative real-valued function.

Using the "partitioning of range of analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png" philosophy, the integral of analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png should be the sum over analysis_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png of the elementary area contained in the thin horizontal strip between analysis_11c971d161e3a32855dca081c99ac51d239a1115.png and analysis_93479b23df7cf877d3bdecca9b3a4207e181399f.png, which is just

analysis_64561d2e90b7acf7a6d1bb28653ebf5d25940d1e.png

Letting

analysis_8b6c5d7fb3a7335c2a880b0962a0e76f46126e59.png

The Lebesgue integral of analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is then defined by

analysis_d2cc8b987115b5eaca1757ef5cf9bf559aa2861b.png

where the integral on the right is an ordinary improper Riemann integral. For the set of measurable functions, this defines the Lebesgue integral.

Measurable function

Let analysis_c48184786640cb39371ca2c31308c6ffed3b24d4.png and analysis_d263068f6ff4f5c92e8c8117813cc69f65d36645.png be measurable spaces.

A function analysis_8fd511771ed24948876570c0e0ef76a099b637c2.png is said to be measurable if the preimage of analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png under analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is in analysis_017948b866be67b1a8e56a6b5f8848f823410c34.png for every analysis_8dc64034b6e33b90a540b84958c46fcfc5c121ec.png, i.e.

analysis_3a4a34795eeb9b31f4209bb8fa01fe9a15d67481.png

Radon measure

  • Hard to find a good notion of measure on a topological space that is compatible with the topology in some sense
  • One way is to define a measure on the Borel set of the topological space

Let analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png be a measure on the sigma-algebra of Borel sets of a Hausdorff topological space analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

  • analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png is called inner regular or tight if, for any Borel set analysis_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png, analysis_db375b73c0e07268ccfddecb004db6b906fb51ad.png is the supremum of analysis_7c257de45927e5350f18c475d8566c8dc366c1d4.png over all compact subsets of analysis_1641d18cc980f8db14cdff95d7417a8526eef446.png of analysis_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png
  • analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png is called outer regular if, for any Borel set analysis_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png, analysis_db375b73c0e07268ccfddecb004db6b906fb51ad.png is the infimum of analysis_1d9db1d5a726ac7f5bb690e7ad6c7c4bd3e9e614.png over all open sets analysis_da9cb51849b13210fa778a80b9907f86fe90c379.png containing analysis_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png
  • analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png is called locally finite if every point of analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png has a neighborhood analysis_da9cb51849b13210fa778a80b9907f86fe90c379.png for which analysis_1d9db1d5a726ac7f5bb690e7ad6c7c4bd3e9e614.png is finite (if analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png is locally finite, then it follows that analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png is finite on compact sets)

The measure analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png is called a Radon measure if it is inner regular and locally finite.

Suppose analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png and analysis_e2697d80120f3dc874ffb73b07cdc68259d6fd48.png are two analysis_026b58b7d0ddfb732bc0a0630d01a61cc001cc27.png measures on a measures space analysis_e7e86caa75e92b35a05bceec215fb7422241feba.png and that analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png is absolutely continuous wrt. analysis_e2697d80120f3dc874ffb73b07cdc68259d6fd48.png.

Then there exists a non-negative, measurable function analysis_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png on analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png such that

analysis_f74de685294b3c880c2838c4bdbb365f7197ec02.png

The function analysis_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png is called the density or Radon-Nikodym derivative of analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png wrt. analysis_e2697d80120f3dc874ffb73b07cdc68259d6fd48.png.

Continuity of measure

Suppose analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png and analysis_e2697d80120f3dc874ffb73b07cdc68259d6fd48.png are two sigma-finite measures on a measure space analysis_f15b78d0d3d5c4397bb64f0f460372309bae5e57.png.

Then we say that analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png is absolutely continuous wrt. analysis_e2697d80120f3dc874ffb73b07cdc68259d6fd48.png if

analysis_a7b0b8568bae9b6a2e1f2e6e474c2ad3166ab962.png

We say that analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png and analysis_e2697d80120f3dc874ffb73b07cdc68259d6fd48.png are equivalent if each measure is absolutely continuous wrt. to the other.

Density

Suppose analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png and analysis_e2697d80120f3dc874ffb73b07cdc68259d6fd48.png are two sigma-finite measures on a measure space analysis_f15b78d0d3d5c4397bb64f0f460372309bae5e57.png and that analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png is absolutely continuous wrt. analysis_e2697d80120f3dc874ffb73b07cdc68259d6fd48.png. Then there exists a non-negative, measurable function analysis_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png on analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png such that

analysis_d9583a6befa91297a06b1a5afb9d158ceb37de3c.png

Measure-preserving transformation

analysis_61e5589f0f948fdfe5922eb14ba15d50dc67c567.png is a measure-preserving transformation is a transformation on the measure-space analysis_85836c3c18ee764fbdef801c3631f37f2456dcf7.png if

analysis_736e98111fbdfac22260d34ea68d36fefccce424.png

Sobolev space

Notation

  • analysis_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png is an open subset of analysis_004097ff73cb85a0f596c8a3b60218ece0e16be1.png
  • analysis_639a884166da121d75367faf8e90ba8f1fcbeea7.png denotes a infinitively differentiable function analysis_078b85cd3478400338e3a1ee425c2a468644be7e.png with compact support
  • analysis_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png is a multi-index of order analysis_3a4189a25e6c0ad201e3ae2a133fdd9e6fffdab0.png, i.e.

    analysis_133c1389abe887c4224b062ad4751431be22d788.png

Definition

Vector space of functions equipped with a norm that is a combination of analysis_5bda34178cc82d1588df2a197af81c5f47d96d87.png norms of the function itself and its derivatoves to a given order.

Intuitively, a Sobolev space is a space of functions with sufficiently many derivatives for some application domain, e.g. PDEs, and equipped with a norm that measures both size and regularity of a function.

The Sobolev space spaces analysis_e82554e3d1082fd9ce05ec7527b38e2f5029f377.png combine the concepts of weak differentiability and Lebesgue norms (i.e. analysis_5bda34178cc82d1588df2a197af81c5f47d96d87.png spaces).

For a proper definition for different cases of dimension of the space analysis_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png, have a look at Wikipedia.

Motivation

Integration by parst yields that for every analysis_38819d1d6c06b8aa737e986909ceae6dc478655b.png where analysis_4028f9016d4f7a233195bc931939fa3360d350cd.png, and for all infinitively differentiable functions with compact support analysis_639a884166da121d75367faf8e90ba8f1fcbeea7.png:

analysis_395ae961a7b4560cd48e4b1e1037e7f73b929f59.png

Observe that LHS only makes sense if we assume analysis_8b0e7a5ea4d56f5afdac3c2658ca87bd50b0ad6f.png to be locally integrable. If there exists a locally integrable function analysis_d198c0d06b0da5525f9a966222601066509ac01d.png, such that

analysis_fbf05489ec587bf169ce19dcf6c3d42f801937a7.png

we call analysis_d198c0d06b0da5525f9a966222601066509ac01d.png the weak analysis_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png -th partial derivative of analysis_8b0e7a5ea4d56f5afdac3c2658ca87bd50b0ad6f.png. If this exists, then it is uniquely defined almost everywhere, and thus it is uniquely determined as an element of a Lebesgue space (i.e. analysis_5bda34178cc82d1588df2a197af81c5f47d96d87.png function space).

On the other hand, if analysis_38819d1d6c06b8aa737e986909ceae6dc478655b.png, then the classical and the weak derivative coincide!

Thus, if analysis_f33b408b8aed4567e7198c9a0bb79e3bbde50e04.png, we may denote it by analysis_b7fdf1ec978c965b42e6e55ef35243184c58a848.png.

Example

analysis_93f302a90840357efcd526fcdcfb4371d1bba5fb.png

is not continuous at zero, and not differentiable at −1, 0, or 1. Yet the function

analysis_08951f8c0ba6006aa8725bcce741fd587043e16a.png

satisfies the definition of being the weak derivative of analysis_d318a5efb8e9c28dd4bb1fead2cc1b6c4684a13f.png, which then qualifies as being in the Sobolev space analysis_81acc4334be481255fc9139da6d9f89a50e19e3d.png (for any allowed analysis_7225b076f6e6326f1636b11d1aad8de58bcc4761.png).

Ergodic Theory

Let analysis_61e5589f0f948fdfe5922eb14ba15d50dc67c567.png be a measure-preserving transformation on a measure space analysis_85836c3c18ee764fbdef801c3631f37f2456dcf7.png with analysis_d5e912600ab8891e4a4e9ce716a36aafd21c0f9a.png, i.e. it's a probability space.

Then analysis_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png is ergodic if for every analysis_0eee8e67cb0b4331b9a6c9f62836d10e5c186781.png we have

analysis_ef05cd277cb9692f4df1dd03297e1ba60bb6f147.png

Limits of sequences

Infinite Series of Real Numbers

Theorems

Abel's formula

Let analysis_edf8753a5a750154d3b2eda69316d0914dae1bc6.png and analysis_38fc79961834e5a23659c13c6cdf67daa82dfabd.png be real sequences, and for each pair of integers analysis_49dd978decd22a11640e762922742a0a3553793b.png set

analysis_221840d8e6869b54e2d104414b518144c186d2dc.png

Then

analysis_70ee71601279ede4f0e355c52b8a6d9aea7935c2.png

for all integeres analysis_6d590463eca4404c6ce717826f87bbba0d82f70d.png.

Since analysis_f73a65e1eeb85f5c8139bdfa9b580a449cb489b0.png for analysis_7b46dc40a80aa60513da2e30da8f8a822c58cacd.png and analysis_5f46949795d31f037e3d957774b6d22fc9fa4912.png, we have

analysis_3a77ae1a71eb0ad4b5b030b391d70a7c7e40d069.png

Infinite Series of Functions

Uniform Convergence

Theorems

Cauchy criterion

Let analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png be a nonempty subset of analysis_492d525117d0dcc93d066c8759f46b98cf9980ca.png, and let analysis_e5acff82b29ef83779f0b8802f221581af6629c7.png be a sequence of functions.

Then analysis_617b87515c58d75e4faeff2a8dbc4b795e512b7c.png converges uniformly on analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png if and only if for every analysis_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png there is an analysis_7c894fec43a3124cb3587458b79302e97436674a.png such that

analysis_c89c0ccb4dd6f951a8ae1916ddbfcbe94e130bf7.png

for all analysis_a75b7783a1f07c2000e3a9ef2d944140b38f6435.png.

Generally about uniform convergence

Let analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png be a nonempty subset of analysis_492d525117d0dcc93d066c8759f46b98cf9980ca.png and let analysis_39629cf44a5442608cb86dd3d49ce7f76bda097d.png be a sequence of real functions defined on analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png

i) Suppose that analysis_6b2017900a7ab6b4f73b76d9ad6382601a66f6d1.png and that each analysis_2d8451421f69781e044825756e911af8aa0ab2f2.png is continuous at analysis_6b2017900a7ab6b4f73b76d9ad6382601a66f6d1.png. If analysis_4bfea4c2a1822082a823fa158ac3b5b6bcc3346f.png converges uniformly on analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png, then analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is continuous at analysis_6b2017900a7ab6b4f73b76d9ad6382601a66f6d1.png

ii) [Term-by-term integration] Suppose that analysis_8f3420acfb5afaf40a82c8488bc62c205289e331.png and that each analysis_2d8451421f69781e044825756e911af8aa0ab2f2.png is integrable on analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png. If analysis_4bfea4c2a1822082a823fa158ac3b5b6bcc3346f.png converges uniformly on analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png, then analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is integrable on analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png and

analysis_4b725415fe802b0bec0ee7c002b88d71d9e15add.png

iii) [Term-by-term differentiation] Suppose that analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png is a bounded, open interval and that each analysis_2d8451421f69781e044825756e911af8aa0ab2f2.png is differentiable on analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png. If analysis_d51cf91a6fdf9189bf6180dd7d520ac6026248d3.png converges at some analysis_6b2017900a7ab6b4f73b76d9ad6382601a66f6d1.png, and analysis_9d54a4f93a0a95c5d9d6da4bb5ae806ab3475864.png converges uniformly on analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png, then analysis_fb006b22bb25b436ad8bec59edadfa5f8e812c8e.png converges uniformly on analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png, analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is differentiable on analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png, and

analysis_d7f9fcc8a89c751eb86e0478ab6e85d7153f7bc8.png

Suppose that analysis_86fff707df410a65631e9a7f612d32ddf7486c48.png uniformly on a closed interval analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png. If each analysis_617b87515c58d75e4faeff2a8dbc4b795e512b7c.png is integrable on analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png, then so is analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png and

analysis_08dbe206aed93dd59273ff79d5638a3892b7e761.png

In fact,

analysis_ab42ca244e2502ba37498d1aa9846a988ba093a4.png

uniformly on analysis_0ced0ad4a59e174c5d4653d36ead7e15dffb0016.png.

Problems

7.2.4

Let

analysis_2f13a86ec969a40963fbb16b2a266c1b6e63e379.png

analysis_0b9664c4f9b6d82d68954cab04fcd7f305908081.png

  1. Show that the series converges on analysis_6c4520a42f92b8385ac5219bc219da8779f218ab.png
  2. We can integrate series term by term

Start by bounding the terms in the sum:

analysis_31adf1cfd751bae6da3ed11e7cf05324bbef382d.png

And since the series analysis_73d8b9c98078bc86646ee3f3fe8664e3a0479994.png converges, the series in question converges.

Further,

analysis_9adab44d4ae0ff99bddb94f7b7b69a283f512850.png

Here we note that the numerator will only take on the values analysis_14e1d572f25955a2407e0f930d21d835f6697411.png, and in the non-zero cases the denominator will be as in the claim.

TODO 7.2.5

analysis_97a36ab3f1175865583cf39d5ab4fd23c9df22c3.png

converges pointwise on analysis_492d525117d0dcc93d066c8759f46b98cf9980ca.png and uniformly on each bounded interval in analysis_492d525117d0dcc93d066c8759f46b98cf9980ca.png to a differentiable function analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png which satisfies

analysis_ed3f32f88d75457296e7cb9c0de397b1ac2955b2.png

for all analysis_b0706adb445a2c20a4d236cf389c6138cf2c5ccb.png

  1. Pointwise convergence on analysis_492d525117d0dcc93d066c8759f46b98cf9980ca.png
  2. Uniform convergence analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png with analysis_ace2bca881e8ad00894c8bc9404c9d32b19d7e07.png

For 1. we observe that

analysis_e2e43ea63e7167f43da2bb492c1e63f281f2b8fa.png

analysis_2a79c4f80a32d33f9f7aded789563fd9e4bab6a1.png

where the last step is due to the sum being a Telescoping series, which equals 1.

We then use the M-test, hence we get convergence in uniform on analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png.

Now that we know that the series converges, we need to establish that the function satisfies the boundaries.

analysis_222e88d0ffa3cf03d5641ae6ac70d59d2a7955e9.png

Where analysis_69689416ea4ea0291c51315b03080e7faa4eb66e.png becomes analysis_88ff129d7bbaa864ac622e34ebba97801e661a0b.png if we can prove that RHS converges.

TODO 7.2.6

analysis_cb1b957c42232fe9d24fe534e3af875f10265538.png

  1. Look at the more general case

analysis_e064b7ddb24267583fa067a89f0a49238f1e8e78.png

  1. Look also at analysis_562e4c81813491489393ccdb3875f72dbcae156d.png
Workshop 2
  • 6
    • Question

      Let analysis_5d0c7356b716bff4f9d8bcccb5fee82e68a7737c.png for analysis_31a3d0292a794fb3cbea1fbccc31638235a1ac7d.png. Prove that analysis_617b87515c58d75e4faeff2a8dbc4b795e512b7c.png converges pointwise on analysis_5cb819dbdaa11557a460fe04a5ef95eede596daa.png and find the limit function. Is the convergence uniform on analysis_5cb819dbdaa11557a460fe04a5ef95eede596daa.png? Is the convergence uniform on analysis_cca855e3b298dbf6e56673409068ea685402d3e9.png with analysis_f669321c14c029084bfaa68010166589e530f2ac.png?

    • Answer

      First observe that for analysis_e862b8e23997ab49cde968039b81be354c606041.png and analysis_49c1e4c835061e1643ceb67c6a764cbd29a8b696.png we have

      analysis_0f1a0cbe75a83c0a9b9d877728a48eb33a36836b.png

      And for analysis_4a3b22922580e4a653364bd8f594a09dc4a793fb.png

      analysis_77ecfce750c9d2cd231bb5511000d95f5e066332.png

      Therefore the limiting function is

      analysis_630cb7c155ef1ea807694ca37ade24c2da4bb269.png

      Is the convergence uniform on analysis_5cb819dbdaa11557a460fe04a5ef95eede596daa.png? No! By Thm. 7.10 in introduction_to_analysis we know that if analysis_86fff707df410a65631e9a7f612d32ddf7486c48.png uniformly then

      analysis_8358cc0385a0db9096755829f95b7c05ca17237c.png

      but in this case

      analysis_fa96029f6a2b592048b6058bfefbb51455107d88.png

      Hence we have a proof by contradiction.

      Is the convergence uniform on analysis_cca855e3b298dbf6e56673409068ea685402d3e9.png? Yes!

      analysis_6a081c25ae703c916aca5c75ebf765262e1c306f.png

      and

      analysis_b234e214d3e7665352c50e1acbace5261e87a00f.png

      hence analysis_86fff707df410a65631e9a7f612d32ddf7486c48.png uniformly on analysis_cca855e3b298dbf6e56673409068ea685402d3e9.png for analysis_f669321c14c029084bfaa68010166589e530f2ac.png.

  • 7
    • Question

      Let analysis_b5e0546ef7407ce9fd3b3136e02b9ed0f182ec2d.png be a sequence of continuous functions which converge uniformly to a function analysis_1b28688a2f600a83d425c4c444313a873bb8c2ae.png. Let analysis_e6cdf4866830b5a5e4b9d91b46225b629e9fc435.png be a sequence of real numbers which converges to analysis_b0706adb445a2c20a4d236cf389c6138cf2c5ccb.png. Show that analysis_b3fed575fe7242d38fe0c0ae800f902a23504e27.png.

    • Answer

      Observe that

      analysis_7a77a0233a598a71fecfb37555a7166f49fbd5e3.png

      for some analysis_284d8afb66b5b4e7be63fa0ac88f370da6c95ebd.png and analysis_1beed402304e57a1708db15afc7a5ca9d7e96483.png. We know

      analysis_66a1d63999f1d04a075bad1d96c50950075b4b1e.png

      and for all analysis_0809b466cd9de765ec30e362bd8646103cb497eb.png

      analysis_271c114b1ed8c0a7263a7884942eb633123039de.png

      Further, by Theorem 7.10 introduction_to_analysis,

      analysis_8470ef2fe57583ca388e24aed1958fa7b6901979.png

      which implies

      analysis_5acdc3735812f803b519d19a256d3d80e75267c8.png

      Therefore, for analysis_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png, let

      analysis_bcd7873ef16fa098dd5c27b37b4c577c9b3591ce.png

      and

      analysis_591c9ea19f1270d837bfd11650a9c677aa3f2cde.png

      then

      analysis_60c43cce666016d1319f5aaec62ed9bd56f56569.png

      as wanted.

Uniform Continuity

Theorems

Suppose analysis_7093b6aa24e036ee131670d97b6fb3ed1de26e0c.png is continuous. Then it is uniformly continuous.

Problems

Workshop 3
  • 5
    • Question

      Let analysis_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png be an open interval in analysis_492d525117d0dcc93d066c8759f46b98cf9980ca.png. Suppose analysis_a6c6bdea4332bae735801f590f991028aa89e3d0.png is differentiable and its derivative analysis_448c2f570fd2e984d07d7ec3e27a2507188cee52.png is bounded on analysis_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png. Prove that analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is uniformly continuous on analysis_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png.

    • Answer

      Suppose

      analysis_4712421a070fede6cd48c69f4a793e0d88fef29d.png

      for analysis_4174cb28db875db895768f049bab0e4942ebfd79.png. Then by Mean Value theorem we have

      analysis_c63dd3fe8f76f6bee9c9497e5b2f076e4b04648d.png

      Therefore,

      analysis_68287dc2935daa21b2dfb2f0fe015140550d56dd.png

      Thus, let

      analysis_31344036fbde7509f2cc517aff48055b06e4b704.png

      Then

      analysis_f30cd1734903954082844afee19f8fa103fc3a40.png

      Hence analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is uniformly continuous.

Power series

Definitions

Let analysis_aab1642aa184d95e27e90808ae243e08a55ea317.png be a sequence of real numbers, and analysis_8e1920707fd0ecf563db5e5cf42dac8c316a64e9.png. A power series is a series of the form

analysis_6a15acda5a2e971f6b60785a00b1452f44282aff.png

The numbers analysis_9c17aa32c24ff4dbc420cd454dec844aef475372.png are called coefficients and the constant analysis_cc6b5c35669c3b0901ef37cf78827ca69227d7b2.png is called the center.

Suppose we have the power series

analysis_6a15acda5a2e971f6b60785a00b1452f44282aff.png

then the radius of convergence analysis_3d136f0fc4860468633907421c098b9feb0eef24.png is given by

analysis_2981167189095141eec53353a81bd3c54c38867b.png

unless analysis_01938648baada593274986ecaa6fe7c152b2cd88.png is bounded for all analysis_4354cc4e2ffec18c80a5a9c861d7985a3b13247f.png , in which case we declare analysis_818f766f555514b7546423d3fe83e74959d04207.png

I.e. analysis_3d136f0fc4860468633907421c098b9feb0eef24.png is the smallest number such that all series with analysis_be2ffff85eeee718237462068d15a1fb4f3e0e4d.png is bounded.

Analytic functions

We say a function is analytic if it can be expressed as a power-series .

More precisely, analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is analytic on analysis_8029cb3f39ae62a353d900e1f63738a99bced91b.png if there is a power series which converges to analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png on analysis_8029cb3f39ae62a353d900e1f63738a99bced91b.png.

Theorems

analysis_410e058c89ed38d538d917ba5f76d6201ec05ae6.png

This holds in general , thus is basically another way of defining the radius of convergence .

analysis_58bb6d233f77668a16123f9d37002966dc3b0564.png

provided this limit exists.

Converges to a continous function

Assume that analysis_b578a2fbbc174a3f08016a404b309fd91ea8d121.png. Suppose that analysis_5a96d2bc3d00056bdb669a07c7f754cb9e27371b.png. Then the power series converges uniformly and absolutely on analysis_c97cb8e737f1427ff7388f2853e3b96ce8c2690a.png to a continuous function analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png. That is,

analysis_36c4c43f0a48c0abe6ee5a8d609966ba0bcc4b38.png

Taylor's Theorem

Suppose the radius of convergence is analysis_3d136f0fc4860468633907421c098b9feb0eef24.png. Then the function

analysis_a8b0ae18bc3dfebfcc2877000aed7a9ee8311a14.png

is infinitely differentiable on analysis_6292bfb2ddecfa01fa24e585b2be24d9d5853a3e.png, and for such x,

analysis_8bed484c7636bf7b77e6b831f4aca2dfdf18c52e.png

and the series converges absolutely, and also uniformly on analysis_c02718209d67e8732fa8391d02d51283e66e2753.png for any $ r < R$. Moreover

analysis_7fb6a3852ee4f2caa0472be9b4f6c366b10eb987.png

Consider the series

analysis_db72d6914e1f87f39f34098df6179319cc5fa31e.png

which has radius of convergence analysis_3d136f0fc4860468633907421c098b9feb0eef24.png and so converges uniformly on analysis_c02718209d67e8732fa8391d02d51283e66e2753.png for any analysis_dc7d9be67bf3970c312eb956670a70e1c1a000b9.png.

Since

analysis_153dba5f64f07fb72af14a29ddef8d0f721bdfbf.png

and analysis_3666d1c11884251e9beabc0a4a4c0411c2182544.png at least converges at one point, then by Theorem 7.14 in Wade's we know that

analysis_fe0a991b1b7cb543534c2dec45d8d81ccdddf47c.png

Further, we have analysis_936b4fc7b943b9be1d15a000ad3670c399804d6e.png and analysis_41f602aa8661753e90aa89dd5efc8e805db8f8c0.png, which we can keep on doing and end up with

analysis_2bb67ad8169b25c003b2a21a78911d29c76d9bfd.png

Problems

Finding radius of convergence

The power series

analysis_ea85ca62ebaa7567e20158dd763e1f5f9f032e16.png

has a radius of convergence analysis_f060e09c051b5076acefac902df29dff110b30ec.png, and converges absolutely on the interval analysis_dffb504dc9a4ba1265d728284d9a9fc8f849e8d2.png.

One can easily see that the series convergences on the the interval analysis_3dfc12a5eb00c886dd23b7f183474477f7b04dc9.png, and so we consider the endpoints of this interval.

  • analysis_c844c7aad5fb88858d9a3133ee0bf72fff903cd1.png convergences
  • analysis_e68ed9fe2278a67699a8b06962e565ef5bb6402d.png, which is known as the harmonic series and is known to diverge

7.3.1

a)

The series

analysis_1639e537690c3b97a28f14a8a704f7e6785b6ad7.png

converges on the interval analysis_dffb504dc9a4ba1265d728284d9a9fc8f849e8d2.png, and has radius of convergence analysis_f060e09c051b5076acefac902df29dff110b30ec.png.

analysis_1639e537690c3b97a28f14a8a704f7e6785b6ad7.png

Letting analysis_2d546d7614d12c73218b1f94d83517e6f2ecee42.png we write

analysis_df27583aba0410ee453bc7efce5166300b3da604.png

Using the Ratio test, gives us

analysis_3d59620e112cc6ad3902fc3c083460ee1da05f7e.png

Thus we have the endpoints analysis_ce90845a3935abdbf95113974191d5f35b358844.png and analysis_4468973182b954eeeb1a22bfe0c5b928511fa9f2.png.

For analysis_49c1e4c835061e1643ceb67c6a764cbd29a8b696.png we get the series

analysis_509bebb29039f878c145c74b9678595f161cd4d3.png

which converges.

For analysis_8e96676855ce43bfa61c9a5d0004af0142ea9dab.png we get the series

analysis_226a638c741338313c4ae868f7f26741f5824ffa.png

for which we can use the alternating series test to show that it converges.

Thus, we have the series converging on the interval analysis_dffb504dc9a4ba1265d728284d9a9fc8f849e8d2.png.

b)

analysis_f2175ebb586ac0630ba9160cae8188a8c59e3f6f.png

We observe that

analysis_926ca6fbdb23db659fdfc4ff22f74c13423cc04d.png

and let analysis_d95eeb85e5939702be59f957e7e629050d93dcfd.png, writing the series as

analysis_db24576f3ff0ed469f4508d3b4d2e664370a4a53.png

and using the root test, we get

analysis_9a9368ed1edc0731ff1128585b8e4a37b2686652.png

where we let the above equal analysis_7bb0bc6108dfa42aab51cbb68fc4765623b52316.png for the sake of convenience.

Then we use the lim-inf definition of radius of convergence

analysis_0c6a733fb4ea639b9d022b9d30fcfe7ca48787ac.png

c)

analysis_bc96c7a1c7dd6eb4c9dfbe3b4172a450d492f105.png

We then let

analysis_7732a453bfab5b45865c19144467c407ea5019e6.png

Then we can apply the root test.

7.3.2

a)

Look at solution to 7.3.1. a)

b)
c)

Integrability on R

Definitions

Partition

Let analysis_ace2bca881e8ad00894c8bc9404c9d32b19d7e07.png with analysis_9c2f16862ccf2498b205c4cb227c1866a266f2c3.png

i) A partition of the interval analysis_51c7bf61624ad32b6b0bfad9b770b15ef3213e5b.png is a set of points analysis_c62904adfe2ebc94ad41338140d68fbfe633d856.png such that

analysis_70472c25ad5b250bc3426a92ad5c1bf535a5fabe.png

ii) The norm of a partition analysis_d1d6eeb3a9dbde6dda8c717122d44f664882ec42.png is the number

analysis_5c578f1758f2b767bede3ef7ae434d4c24a1bc46.png

iii) A refinement of a partition analysis_d1d6eeb3a9dbde6dda8c717122d44f664882ec42.png is a partition analysis_9fe46e6ccfd065fb50f2eeb9cbdc65835e9ec321.png of analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png which satisfies analysis_390f34e5e3142d1bc291373a59c8d10d2a49b3d5.png. In this case we say that analysis_9fe46e6ccfd065fb50f2eeb9cbdc65835e9ec321.png is finer than analysis_a9090c77ce9916955c745920bf8f134a0932d59d.png.

Riemann sum

Let analysis_ace2bca881e8ad00894c8bc9404c9d32b19d7e07.png with analysis_9c2f16862ccf2498b205c4cb227c1866a266f2c3.png, let analysis_37b452c3dab6cb4da24c54e7069a1c0dabfd4c72.png be a partition of the interval analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png, set analysis_291839531087b3b6cebf4f804dc2377eb7c66fa8.png for analysis_1ae091ca5feb91759a1023ff74ed228fd01b10f1.png and suppose that analysis_b6328f18afb8b265ac01286bd47c4848a4bec7b9.png is bounded.

i) The upper Riemann/Darboux sum of analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png over analysis_a9090c77ce9916955c745920bf8f134a0932d59d.png is the number

analysis_20884c1afb19d41430d81843ab7fc02b0ed1c270.png

where

analysis_b5d45f4cf97fd57b37c3ec7877a98e535952e512.png

ii) The lower Riemann/Darboux sum of analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png over analysis_a9090c77ce9916955c745920bf8f134a0932d59d.png is the number

analysis_d708b2e5b76bdae6faeae60a7337dffdf299dca3.png

where

analysis_f4e7b8b26aa14f2756864afee81f2593a0f37e13.png

Since we assumed analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png to be bounded, the numbers analysis_14ac1f8ff481a6a62729dc3f4106c604ca2e7ef7.png and analysis_93c2efe4f5ef01382c92c08b1380b54ed7bd4864.png exist and are finite.

Riemann integrable

Let analysis_ace2bca881e8ad00894c8bc9404c9d32b19d7e07.png with analysis_9c2f16862ccf2498b205c4cb227c1866a266f2c3.png. A function analysis_7093b6aa24e036ee131670d97b6fb3ed1de26e0c.png is said to be Riemann integrable on analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png if and only if analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is bounded on analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png, and for every analysis_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png

analysis_968439ef652044491ff0d9940329b7157447dee0.png

I.e. the upper and lower Riemann / Darboux sums has to converge to the same value.

Riemann integral

Let analysis_ace2bca881e8ad00894c8bc9404c9d32b19d7e07.png with analysis_9c2f16862ccf2498b205c4cb227c1866a266f2c3.png, and analysis_b6328f18afb8b265ac01286bd47c4848a4bec7b9.png be bounded.

i) The upper integral of analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png on analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png is the number

analysis_0682ea7fb1f55880d7f38c464ea0053c07b96362.png

ii) The lower integral of analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png on analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png is the number

analysis_4ed84f928af0a9d0e9e7a6c3047a3cb1a506d770.png

iii) If the upper and lower integral are equal, we define the integral to be this number

analysis_b64b7bde8efb4d9a2eabbf0e53cbe7a5613789b8.png

The following definition of the Riemann sum can be proven to be equivalent of the upper and lower integrals using introduction_to_analysis.

Let analysis_b6328f18afb8b265ac01286bd47c4848a4bec7b9.png

i) A Riemann sum of analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png wrt. a partition analysis_60fe63cc43b53a17c0104747c2e07800708f8a0e.png of analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png generated by samples analysis_8410f0d55c1e0313be4d80ea0f39ac657710c3fc.png is a sum

analysis_505ee592d62f5090660a2cf6065933c9fb4d393e.png

ii) The Riemann sums of analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png are said to converge to analysis_d16653d46f734308b81e002df94a41646c5538d9.png as analysis_4dd182ed08c3df0a98f506ebc6a2a28669962de2.png if and only if given analysis_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png there is a partition analysis_6f8307982ed302f39985df1599d6dd730d5c9a39.png of analysis_dec77c63616388aa91bdbd7078119ab733f04822.png such that

analysis_6f14ecfb6c420b5e2eead90bc7bdd0891e29dd69.png

for all choices of analysis_3dc15bc7c2a0b8627fc879aeab4c32e20cf01af5.png. In this case we shall use the notation

analysis_d8b36464c9dd29d0a0dfbf87e88cb3d6869632f6.png

analysis_00b67c75331044cf0a69c738528f1b76d7c1796a.png is just some arbitrary number in the given interval, e.g. one could set analysis_a9660a5ab2450513f2a745dfdbf3decb4cb3dd60.png.

Theorems

Suppose analysis_ace2bca881e8ad00894c8bc9404c9d32b19d7e07.png with analysis_9c2f16862ccf2498b205c4cb227c1866a266f2c3.png. If analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is continuous on the interval analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png, then analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is integrable on analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png.

Telescoping

If analysis_76e3d6bb245dd0de74074f7f0f96248599bc58d4.png, then

analysis_df87e284ad83bd615cda94c3133db267201b8371.png

This is more of a remark which is very useful for proofs involving Riemann sums, since we can write

analysis_1714bfac792b8ab1e14b6375decc6772060614b1.png

This allows us to write the following inequality for the upper and lower Riemann / Darbaux:

analysis_ac36ce3dc90fde5c54210255b9491bc258376b65.png

in which case all we need to prove for analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png to be Riemann integrable is that this goes to zero as analysis_d5568e6b71fbba5051b9a91c0253e8cdee6c6c98.png, or equiv. as we get a finer partition.

Mean Value Theorem for Integrals

Suppose that analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png and analysis_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png are integrable on analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png with analysis_baf93323f41df004d1df51568c759e1de81da30b.png for all analysis_0ced0ad4a59e174c5d4653d36ead7e15dffb0016.png. If

analysis_f8f4b3c88c7ce6cab38676c5370badc5eb758b0d.png

then

analysis_2f9e4fe36f2038427fdb1c15b4266b712646dc53.png

In particular, if analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is continuous on analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png, then

analysis_5a1dbccd7fdf1d2c5c3f9bfb3a654cd89d6f423d.png

introduction_to_analysis

Suppose that analysis_198a64800b20de12f4aafa400d8ff0f00fd78fb2.png are integrable on analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png, that analysis_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png is nonnegative on analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png, and that

analysis_31641137a2d210521df53fd5eb7aa52cdd199fd1.png

Then there is an analysis_6507257ac52e6d535062474fde37b9d964a028e3.png s.t.

analysis_c6ed07370391769cba8d09a9a82de9ac757f2f30.png

In particular, if analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is also nonnegative on analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png, then there is an analysis_6507257ac52e6d535062474fde37b9d964a028e3.png which satisifies

analysis_6675a26a29e76591c1243d17305e3d4d1ebe1751.png

introduction_to_analysis

Let analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png be a real-valued function which is continuous on the closed interval analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png, analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png must attain its maximum and minimum at least once. That is,

analysis_207540d3b13f019d564270137124e3e3bebd3395.png

Fundamental Theorem of Calculus

Let analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png be non-degenerate and suppose that analysis_b6328f18afb8b265ac01286bd47c4848a4bec7b9.png.

i) If analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is continuous on analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png and analysis_62bd535e056770ae5f1bd18410aa7bbd735c2248.png, then analysis_d070dbff13b683b6ebafa2f58291f96264bb8eff.png and

analysis_178876523ea76a2725b051914b909be228bfa0d0.png

ii) If analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is differentiable on analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png and analysis_448c2f570fd2e984d07d7ec3e27a2507188cee52.png is integrable on analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png then

analysis_c60926f4fcec25e33d98d8a3d56c7cd3396ea8fb.png

introduction_to_analysis

Non-degenerate interval means that analysis_9d12846ee5476c4c275a3f27979b84d3b93eb1c9.png.

Integrability on R (alternative)

Let analysis_f9bf6d664d9d2a949a6d9944de07b272a08dcb57.png. Then we define the characteristic function

analysis_3a51ff2b2b1ff8c84645cfc018fde8054a0c1188.png

Let analysis_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png be a bounded interval, then we define the integral of analysis_234c2ccac8379b0014c97e22f0851cb77f88e618.png as

analysis_63a2ea5d3cbeeaefabba50cceb2851062e8058ef.png

Reminds you about Lebesgue-integral, dunnit?

We say analysis_b201d8df50ba492a0e91c8dd51dee9fdbe27f6f3.png is a step function if there exist real numbers

analysis_d37e3a2ce0dc61149642fe157b2617553e11492c.png

such that

  1. analysis_ad6edee32f00f6fee72e111d3a49665325170036.png for analysis_668659ad2b9ccd5bc2009616e88b49b0b6c5a0d8.png and analysis_e39dc516d751a46e7b41e782ed542b03e6ccf213.png
  2. analysis_d21892f3bdfae9ec08a78f3061484c467c1030ec.png is constant on analysis_bc21b0e1281684b173d4704fb8498ba065f87c07.png

We will use the phrase "analysis_d21892f3bdfae9ec08a78f3061484c467c1030ec.png is a step function wrt. analysis_b9995cad286b4a42b041c675671d1d1e0200b84b.png" to describe this situation.

In other words, analysis_d21892f3bdfae9ec08a78f3061484c467c1030ec.png is step function wrt. analysis_b9995cad286b4a42b041c675671d1d1e0200b84b.png iff there exists analysis_e36a491e354bf439e6231572d599941a33701f16.png such that

analysis_3377f4bf6d0b03fea70756085687d7d36652c398.png

for analysis_7492197345418252123e11b9734982d67db2ef51.png.

If analysis_d21892f3bdfae9ec08a78f3061484c467c1030ec.png is a step function wrt. analysis_b9995cad286b4a42b041c675671d1d1e0200b84b.png which takes the value analysis_0d4cf74d4d6c3a44f3d67f205f866ba36ea8d4aa.png on analysis_1bf77781868af6b9d8845262aaf25f2e57a4b1e2.png, then

analysis_eeafa1d940ea805d1c03e53f3bf81ee40a8e093c.png

Notice that the values analysis_9e5a9db635c4210e1047b4e3232aa0bbd407a7b9.png have no effect on the value of analysis_1d0ec2c3af6235e239713373cd8e870ce443de21.png, as one would expect.

Let analysis_1b28688a2f600a83d425c4c444313a873bb8c2ae.png. We say that analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is Riemann integrable if for every analysis_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png there exist step functions analysis_d21892f3bdfae9ec08a78f3061484c467c1030ec.png and analysis_d21892f3bdfae9ec08a78f3061484c467c1030ec.png such that

analysis_3b6b9be0a8ef20702f80c850db6fd4278f7c4b40.png

and

analysis_79536e6f72e1cb1f05b3fdb773f7c89ae35b4316.png

A function analysis_1b28688a2f600a83d425c4c444313a873bb8c2ae.png is Riemann integrable if and only if

analysis_1b8b616311b0f369aa70b18f0e0e01a290357ff3.png

If analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is Riemann integrable we define the integral of analysis_5c63d5ea2a22b5f130680d30426ca4ffc0f2e453.png as the common value

analysis_9a9049beaccbd61bc404fe3afc0ed0a0ed6896ca.png

or equivalently,

analysis_7ee4b0f5ec40d188ac1da8057127536fa4621a2d.png

A function analysis_1b28688a2f600a83d425c4c444313a873bb8c2ae.png is Riemann integrable if and only if there exist sequences of step functions analysis_1a5e8cc664d575dc9783f32b7f6803f5647c444b.png and analysis_2b9bc6bf51e9a0cd482fd7e739f12dd94e44eab7.png such that

analysis_d602fc1b540a27416681cc526664006614c8c4a7.png

and

analysis_08fa9bf0645317ac982675d29c449ff0e8ed680f.png

If analysis_1a5e8cc664d575dc9783f32b7f6803f5647c444b.png and analysis_2b9bc6bf51e9a0cd482fd7e739f12dd94e44eab7.png are any sequences of step functions satisfying the above, then

analysis_07adb6835edf185a078124d2934c6c4be6b3be5f.png

as analysis_2531823f4e49c847dd9a34d97de0ea57802b96fa.png.

Suppose analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png and analysis_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png are Riemann integrable, and analysis_af43c0b082122b7a7ace41e7204e88f6b1395cf6.png. Then

  1. analysis_508b94f8525bf48d0101018f31fb597847fcda76.png is Riemann-integrable and

    analysis_da5280e88e48370d6e11f5a26f9ea57f02349189.png

  2. If analysis_75de054d7bb331e8c651820f1666490e03120ef3.png then analysis_1b4f438ba37348280ab90075ab34349257b38501.png. Further,

    analysis_aad0db28c3182d7d4fc37868d64160af5a3eec11.png

  3. analysis_a5726c5ecf917b75af3d8909a408349be1f4b347.png is Riemann integrable and

    analysis_44955fbb8787687ee196ab09fbc547f14633f0b6.png

  4. analysis_ccaf436834382b414a32a13b9f03521d6f6d493d.png and analysis_c5c1dfcdb9fdda19c3a63e174522aad8ffcae81f.png are Riemann integrable
  5. analysis_0bf126e4d8e11508a87efffc8c1bb3f94b9d72ee.png is Riemann integrable

Integrals and uniform limits of sequences and series of functions

Suppose analysis_b5e0546ef7407ce9fd3b3136e02b9ed0f182ec2d.png is a sequence of Riemann integrable functions which

analysis_d109e6b3b6dc97b29f2e71d3c8d9304d86232042.png

uniformly.

Suppose that analysis_617b87515c58d75e4faeff2a8dbc4b795e512b7c.png and analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png are zero outside some common interval analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png. Then analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is Riemann integrable and

analysis_6ba6d4b1cd0307c99183d96f551f2f4793c7261f.png

Suppose analysis_1e897c6c0b63b1e720250f613cecaa977f05b71d.png is a non-negative sequence of numbers and analysis_07f6f026f074142fdee0806e48c2eae5ab72b0d6.png is a function that

  1. For some analysis_1641d18cc980f8db14cdff95d7417a8526eef446.png and analysis_0d6385680f9f50537637e277cd21d9caef3c8836.png

    analysis_3c2553635574b858170b97b0d753f99a3d703732.png

  2. For analysis_f2395584a10e97964d805bd4712900ae2e97306d.png we have

    analysis_4b66d58ef9efae95dbd0d706e82599c267f83b68.png

Then

analysis_c2286143e20bf6b2dd034cd71a49c20f79fe89e3.png

for some analysis_4e640a760fb96706b6878acf9d88cb0fdc61d9ee.png.

Problems

Workshop 5

5
  • Question

    Suppose analysis_1b28688a2f600a83d425c4c444313a873bb8c2ae.png is Riemann integrable, and that analysis_6a3483facd6a491101c75a7860766f9e133e5617.png outside of analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png where analysis_9c2f16862ccf2498b205c4cb227c1866a266f2c3.png.

    Show that

    analysis_410bed049a56a0eb9db3c9f9c1159f8dec91d328.png

    is also Riemann integrable.

  • Answer

    analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is Riemann integrable, then there exists step-functions analysis_d21892f3bdfae9ec08a78f3061484c467c1030ec.png and analysis_541a8541573e130dc310c688d0524f6e8fc0a5e3.png such that

    analysis_6eb75726c86421910b6198179fb2edfe92bacbb2.png

    Or rather, for all analysis_0ed882168b2ef3d0f25091a4c007f35cb4c7bb43.png, there exists analysis_e0176b102e4bbaa8dbf64442f3eb5d59de937a1e.png such that

    analysis_779b61f7c3d9c64f3b483714a8dcba017a6611e5.png

    Since analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is integrable on a bounded and closed interval, then analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is bounded and has bounded support. That is

    analysis_6fb1cfc84f91459768766fc4f69c8cba1bce3550.png

    Therefore, by the Mean Value theorem, we have

    analysis_7d006d7a0f9ceecde92af41ed7f1e2da84e86245.png

    Therefore the "integral sum" for analysis_1ba309b89f115a0bf9d3d3b14e7f769c39a44bde.png satisfy the following inequality

    analysis_1bffd66de4b4b7d2fffd7ea8180d83f263a673db.png

    Therefore, for analysis_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png, we choose

    analysis_5b93f11fafaa1589ddd8776214b865a4169d7610.png

    which gives us

    analysis_d087a754991cc94b842a4c568625aa49ae07ea0a.png

    That is, analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is Riemann integrable implies analysis_1ba309b89f115a0bf9d3d3b14e7f769c39a44bde.png is Riemann integrable.

    We've left out the analysis_e7d16c8f48c97a9236fa45c44c17cd03ed310067.png in all the expressions above for brevity, but they ought to be included in a proper treatment.

Metric spaces

Definitions

Metric

A metric space is a set analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png together with a function analysis_2fb20a81453ae51b834895c36c0fc252d0283940.png (called the metric of analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png) which satisfies the following properties for all analysis_f3767362ab45edafa6a1cdb5e277aa24ec258733.png:

analysis_78900306aa42d4837329621261852d25b119a9be.png

Metric space

A metric space is a set analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png together with a function analysis_2fb20a81453ae51b834895c36c0fc252d0283940.png (called the metric of analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png) which satisfies the following properties for all analysis_f3767362ab45edafa6a1cdb5e277aa24ec258733.png:

analysis_78900306aa42d4837329621261852d25b119a9be.png

Balls

Let analysis_f4bda5a2a8228d16ebc8beabfd375043e905ee99.png and analysis_a1805c342197ad3dbf7cfae004af60962120c37f.png. The open ball (in analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png) with center analysis_3778a663528a3e4db5ce811ed0c6cd9f5831579f.png and radius analysis_17bffe02d589bbbc96f6efbdbbdd12efb5d9607b.png is the set

analysis_874a952910ded8ce849a379613f68a8bfe0e5354.png

Let analysis_f4bda5a2a8228d16ebc8beabfd375043e905ee99.png and analysis_a1805c342197ad3dbf7cfae004af60962120c37f.png. The closed ball (in analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png) with center analysis_3778a663528a3e4db5ce811ed0c6cd9f5831579f.png and radius analysis_17bffe02d589bbbc96f6efbdbbdd12efb5d9607b.png is the set

analysis_c10b6c4f01b854fd9766438d0f773c321df80569.png

Equivalence of metrics

We say two metrics analysis_f4e5566de1c75b7700b562b91a80d2308cc66d12.png and analysis_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png on a set analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png are strongly equivalent if and only if

analysis_f1ca5236c2623dc9a9805d35b552ed3df3844c40.png

We say two metrics analysis_f4e5566de1c75b7700b562b91a80d2308cc66d12.png and analysis_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png on a set analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png are equivalent if and only if for every analysis_4c91032750887d8dca229b906727af58e9268232.png and every analysis_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png there exists analysis_0809b466cd9de765ec30e362bd8646103cb497eb.png such that

analysis_5bb2ea58d119f017d22deb12b922c0a4284597fe.png

Closedness and openness

A set analysis_2994ad018dfcbe488dbebb8f1c7dc13f600ddf44.png is said to be open if and only if for every analysis_3b815a7c70ccfac0b1b2050bf7bf8472b528180d.png there is an analysis_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png such that the open ball analysis_3804f1eafe72c11487a1f5e07b676e76956c40b6.png is contained in analysis_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png.

A set analysis_ed7f570bd5e8cada8ea7e894b3dfc90eebb1847a.png is said to be closed if and only if analysis_bb7b62c6f4cda25f4a31cc2e00b60235169524e8.png is open.

Closure and interior

For analysis_3bc94bf0d430e84e44d43e627cd2d129fbd5aef3.png

analysis_b4c66555cdd28b05104461790d127dfb34836292.png

is the interior of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png; it is the largest subset of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png which is open.

Or equivalently, the interior of a subset analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png of points of a topological space analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png consists of all points of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png that do not belong to the boundary of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png.

A point that is in the interior of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is an interior point of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png.

For analysis_3bc94bf0d430e84e44d43e627cd2d129fbd5aef3.png

analysis_38c6fbc1fc76365b0ac2eb13c25b425c9581659d.png

is the closure of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png; it is the smallest set containing analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png which is closed .

Or, equivalently, the closure of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is the union of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png and all its limit points (points "arbitrarily close to analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png"):

analysis_2ebe1cadb5cc77ceb38763f603ea63adc37c0cdd.png

For analysis_3bc94bf0d430e84e44d43e627cd2d129fbd5aef3.png

analysis_8e0cf1e58dc3aabaa83d44a483439e85f5243fe8.png

is the boundary of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png.

Convergence, Cauchy sequences and completeness

Most theorems and definitions used for sequences are readily generalized to metric spaces.

We say a metric space is complete if and only if every Cauchy sequence in analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png converges.

In a metric space analysis_f7eec20146ea61ba7ca25392fa7ea5297cb502ce.png, a sequence analysis_42bc3cb404b9870b4d56e33232762a32f814f3e4.png with analysis_6a39f7b48d3fecf222e8e527ab907f1b3940b2f0.png is bounded if there exists some ball analysis_e38a9a2ceeedc5371ca8a10157cbbcffcdc67853.png such that analysis_dfd5cdf0bc07475c40d384114e2fe011ea264e21.png for all analysis_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png.

In a metric space analysis_f7eec20146ea61ba7ca25392fa7ea5297cb502ce.png, a sequence analysis_42bc3cb404b9870b4d56e33232762a32f814f3e4.png with analysis_6a39f7b48d3fecf222e8e527ab907f1b3940b2f0.png is a Cauchy sequence iff for every analysis_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png,

analysis_c3b8c96716baa417e7550a3fa2d755e38edeba6d.png

Let analysis_f7eec20146ea61ba7ca25392fa7ea5297cb502ce.png be a metric space, then analysis_f7eec20146ea61ba7ca25392fa7ea5297cb502ce.png is said to satisfy the Bolzano-Weierstrass Property iff every bounded sequence analysis_6a39f7b48d3fecf222e8e527ab907f1b3940b2f0.png has a convergent subsequence .

Closedness, limit points, cluster points and completeness

analysis_4c91032750887d8dca229b906727af58e9268232.png is a limit point for analysis_3bc94bf0d430e84e44d43e627cd2d129fbd5aef3.png if and only if there is a sequence analysis_6cf668b05909f20d54793e41f7fd189b1e3e4f7d.png such that analysis_97621c3cd88296433cd36d03df345c6772bc32c9.png as analysis_2531823f4e49c847dd9a34d97de0ea57802b96fa.png.

analysis_4c91032750887d8dca229b906727af58e9268232.png is a cluster point for analysis_3bc94bf0d430e84e44d43e627cd2d129fbd5aef3.png if and only if every open ball centred at analysis_7a84c9a383f9772338016d101ccc096be06af784.png contains infinitely many points of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png.

The following statements are equivalent:

  • analysis_4c91032750887d8dca229b906727af58e9268232.png s a cluster point for analysis_3bc94bf0d430e84e44d43e627cd2d129fbd5aef3.png
  • for all analysis_a1805c342197ad3dbf7cfae004af60962120c37f.png, analysis_28bf535fb19b4ae14e8207de24b4a7e15a43eac3.png contains a point of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png
  • analysis_d96a83b550262469058d03b6ccd2e1835e55b754.png, with analysis_cd76bb1db16c4ec3dd69df1d203f6064825e493d.png for all analysis_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png, s.t. analysis_97621c3cd88296433cd36d03df345c6772bc32c9.png as analysis_2531823f4e49c847dd9a34d97de0ea57802b96fa.png

Every cluster point for analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is a limit point for analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png. But analysis_7a84c9a383f9772338016d101ccc096be06af784.png can be a limit point for analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png without being a cluster point .

A closed subset of a complete metric space is complete

A complete subset of any metric space is closed.

Every convergent sequence is Cauchy, but the opposite is not necessarily true.

Compactness

Let analysis_afd21c6b6c40353c2b9e7dade10c4acda660d515.png be a collection of subsets of a metric space analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and suppose that analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png is a subset of analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

analysis_040fb6dc800d9bfdad95f2d7e7583208d70225d5.png is said to cover analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png if and only if

analysis_2b905407958ef2abb2f5d9cbb5063c0538ebb80f.png

analysis_040fb6dc800d9bfdad95f2d7e7583208d70225d5.png is said to be an open covering of analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png iff analysis_040fb6dc800d9bfdad95f2d7e7583208d70225d5.png covers analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png and each analysis_6e4588b08b22bea71babfa16ad8382b824f40aaf.png is open.

Let analysis_040fb6dc800d9bfdad95f2d7e7583208d70225d5.png be a covering of analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png.

analysis_040fb6dc800d9bfdad95f2d7e7583208d70225d5.png is said to a finite (respectively, countable ) subcovering iff there is a finite (respectively, countable) subset analysis_5dc599b052bbf6ae436775d342162b7815abc2e8.png of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png s.t. analysis_6222e3ee53c281d2664259a74c9de70ff6be1614.png covers analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png.

Let analysis_f7eec20146ea61ba7ca25392fa7ea5297cb502ce.png be a metric space.

A subset analysis_ed7f570bd5e8cada8ea7e894b3dfc90eebb1847a.png is compact iff for every open cover analysis_58e0ee166f3bc784e9e83edf416787a2c2eda6d8.png of analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png, there is a finite subcover analysis_8f03f1e12d2f1e5636a57d9ed1266caf44316445.png of analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png.

I often find myself wondering "what's so cool about this compactness?! It shows up everywhere, but why?"

Well, mainly it's just a smallest-denominator of a lot of nice properties we can deduce about a metric space. Also, one could imagine compactness being important since the basis building blocks of Topology is in fact open sets, and so by saying that any open cover has a finite open subcover, we're saying it can be described using "finite topological constructs". But honestly, I'm still not sure about all of this :)

One very interesting theorem which relies on compactness is Stone-Weierstrass theorem, which allows us to show that for example polynomials are dense in the space of continous functions! Suuuuper-important when we want to create an approximating function.

Let analysis_f539b53d308b3ff1c2d34c50721e22494ea6449a.png be a metric space and let analysis_04dbc02b1b2d96a53221dfd08c2e2d91d8a79104.png. Then analysis_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png is said to be dense in analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png if for every analysis_f4bda5a2a8228d16ebc8beabfd375043e905ee99.png and for every analysis_a1805c342197ad3dbf7cfae004af60962120c37f.png we have that analysis_985895ae6c89d75831334e9926e223d77224f1fc.png i.e., every open ball in analysis_f7eec20146ea61ba7ca25392fa7ea5297cb502ce.png contains a point of analysis_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png.

Or, alternatively, as described in thm:dense-iff-closure-eq-superset:

analysis_9ceb13936a79ddada413bf91c15025210182d59e.png

A metric space analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is said to be separable iff it contains a countable dense subset.

Where with countable dense subset we simply mean a dense subset which is countable.

We say the metric space analysis_f7eec20146ea61ba7ca25392fa7ea5297cb502ce.png is a precompact metric space if for every analysis_a1805c342197ad3dbf7cfae004af60962120c37f.png there is a cover of analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png by finitely many closed balls of the form

analysis_d5b1a553b76011c6df0a3763f5528eb5c14fbff9.png

Let analysis_f7eec20146ea61ba7ca25392fa7ea5297cb502ce.png be a complete metric space and precompact, then analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is compact.

Let analysis_f7eec20146ea61ba7ca25392fa7ea5297cb502ce.png be a metric space. Then analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is said to be sequentially compact if and only if every sequence in analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png has a convergent subsequence.

Let analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a topological space, and analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png a subspace.

Then analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is compact (as a topological space with subspace topology) if and only if every cover of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png by open subsets of analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png has a finite subcover.

If analysis_d7706077e20dc3bdc00ccd5dd86babcd0cdcb73c.png for analysis_772f6c94a850182352f20151b1f1f16671ab0c16.png open in analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png (subspace topology), then analysis_5e57c589063a441981ec7be7b4d029eae6657acf.png open in analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png s.t. analysis_99818225a3ba2cca91c2afc987c38c98200f78d1.png.

Therefore

analysis_82fd669abba42f01eaf048a8a43a3b3f8718bd7a.png

analysis_f9a37dc730cf12c09ef495ac3f5c8057b90caf64.png: Choose finite subcover analysis_9d736408d4b6def73c2a747a1448709919ccc716.png . Then analysis_8046fb4f14734a6b09c98b263b13abdd64728346.png is a finite subcover of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png.

analysis_c5391e19946e07a40e9ce31018090746dcab5e8e.png: Let analysis_097bfdbb1786996adb49c2d288514229f3c0f4a0.png, then analysis_68ecc1fd1c0893a2f2a70074f7fd9b3f0c873c6d.png open in analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png. So we let

analysis_1002777d8e4a0d07649b9be412e2ef25db566849.png

so there exists finite analysis_bfd5dd3891851206850bd1171cb30e55439467d8.png with

analysis_231ef8d9eae4f064ad912e379f0f7cfff61ff3c2.png

Every space with cofinite topology is compact.

Let analysis_cb7261f22f3e6fccff16b4b719e8d0f0482955a6.png. Take some analysis_b9e6a7b79efda48da301677abbf8df89e7321553.png so that analysis_2431672a6205f46a58da5202cc3da6fd46078791.png is finite.

Then analysis_2066e4997ede5eb9315c693dc2150a4e78439789.png, there exists analysis_66a55e82d21574e901c18c3dcfd5de6770cab41c.png in cover with analysis_8f03e1f32a0ce7b2b6732683bb9faba68d63c7e9.png. Therefore

analysis_9d55172afaea9b18f3d19a2ec1867ac759c78200.png

is a finite cover.

Idea: take away one cover → left with finitely many points → we good.

Motivation

Let analysis_39852e5ec4f953f23fa3efc2c6557ea8a7e7114f.png. For which analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png must analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png be bounded?

  • analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png finite
  • If analysis_86306725eb2f816b291310b3040964c1cc9f7a52.png of opens with analysis_98deb3f7a3b82c32748908748d6d6cbd14db19a6.png bounded then analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is bounded.
  • Any continuous analysis_39852e5ec4f953f23fa3efc2c6557ea8a7e7114f.png is locally bounded

    analysis_bd97b4ef47b4d660012ae929231d26b84b0afd27.png

    with analysis_173176958ffc1a113ba78719cd185a02098176d7.png bounded, e.g.

    analysis_3ffd11a5cf7575481569f31471660cf4e5849dfa.png

  • If there exists finitely many analysis_99ca1aaf28e63b863bb92d12f94a36fcba2007bd.png as above, with

    analysis_71f28c0173c06f49dcf89614a2810914c019492b.png

    then analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is bounded.

Compactness NOT equivalent to:
  1. analysis_e68f3a7dab610896c901934a091444ec698d2a55.png is a cover of analysis_492d525117d0dcc93d066c8759f46b98cf9980ca.png
  2. analysis_a917c6146b4dda37c66956a32889e729da8a913a.png covers analysis_5cb819dbdaa11557a460fe04a5ef95eede596daa.png but not finite.
    • analysis_547dc2264649729065dc2513e00cb7127f3b9977.png cover analysis_5cb819dbdaa11557a460fe04a5ef95eede596daa.png → clearly not finite mate
  3. Same as 2, but take finitely many analysis_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png
  4. Follows from 2 by taking subcover to be the whole cover.
  5. analysis_52704b22f154e86d8025bce989539891bd272d77.png always covers analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and has finite subcover (e.g. analysis_21ea047b774d42db2f365803668a8d0d746de541.png)
Examples
  • Non-compact
    1. analysis_cafc1accf62df9dee2bf6897eab4e702f7faaa66.png, so analysis_4dfda54c60f320c3cbcc3949d2fb0e37cdf672d7.png has no finite subcover, so analysis_492d525117d0dcc93d066c8759f46b98cf9980ca.png not compact
    2. Infinite discrete space is not compact. Consider analysis_879dc795ed34bdf0f60b5e86d25bac2d5c14857a.png which is an open cover, but has no finite subcover.
  • Compact
    1. analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png indiscrete so analysis_1bad49a2cd03d1d659541b9ab7d0c7ca14371b31.png. Only open covers are analysis_52704b22f154e86d8025bce989539891bd272d77.png and analysis_71bc4a4827450e36b5dbe6ac39c185928983c7bc.png, and analysis_52704b22f154e86d8025bce989539891bd272d77.png is a finite subcover, hence analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is compact.
    2. Any finite space is compact (for any topology)

Some specific spaces

Banach space
Hilbert space

A Hilbert space analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png is a vector space equipped by an inner product such that the norm induced by the inner product

analysis_a133c1a13236543faa93aa72528a29c33cb7ed44.png

turns analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png into a complete metric space.

A Hilbert space is thus an instance of a Banach space where we specifically define the metric as the square-root of the inner product.

Reproducing Kernel Hilbert Space

Here we only discuss the construction of Reproducing Kernel Hilbert Spaces on the reals, but the results can easily be extended to complex-valued functions too.

Let analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be an arbitrary set and analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png a Hilbert space of real-valued functions on analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png. The evaluation functional over the Hilbert space of functions analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png is a linear functional that evaluates each function at a point analysis_7a84c9a383f9772338016d101ccc096be06af784.png,

analysis_98d8ce8c739e4973e8f0be34a441bbf6e3df107c.png

We say that analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png is reproducing kernel Hilbert space if, for all analysis_7a84c9a383f9772338016d101ccc096be06af784.png in analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png, analysis_c997edc21b0215b223cd458de61b51a474e34761.png is continuous at any analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png in analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png, or, equiavelently, if analysis_c997edc21b0215b223cd458de61b51a474e34761.png is a bounded operator on analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png, i.e. there exists some analysis_4174cb28db875db895768f049bab0e4942ebfd79.png such that

analysis_5fd543a6dc979ccedca03caecfba71382ede04c6.png

While this property for analysis_c997edc21b0215b223cd458de61b51a474e34761.png ensure both the existence of an inner product and the evaluation of every function in analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png at every point in the domain. It does not lend itself to easy application in practice.

A more intuitive definition of the RKHS can be obtained by observing that this property guarantees that the evaluation functional analysis_c997edc21b0215b223cd458de61b51a474e34761.png can be represented by taking the inner product of analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png with a function analysis_d77817a3c4b3732fe8b12ac5f24fc3cbf63cb6c9.png in analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png. This function is the so-called reproducing kernel of the Hilbert space analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png from which the RKHS takes its name.

The Riesz representation theorem implies that for all analysis_4c91032750887d8dca229b906727af58e9268232.png there exists a unique element analysis_d77817a3c4b3732fe8b12ac5f24fc3cbf63cb6c9.png of analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png with the reproducing property

analysis_d460141fa6c59bee119b5bbf7aa2649173942eb1.png

Since analysis_d77817a3c4b3732fe8b12ac5f24fc3cbf63cb6c9.png is itself a function in analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png, it holds that for every analysis_eb83d466c7d035356e9f39998f357cee73da1e26.png in analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png there exists a analysis_402caa1176b0c37ca671b7e19bbcd49336bbab89.png s.t.

analysis_94a1115a9ff0c5ba999ee9dc712bf7bdece71210.png

This allows us to define the reproducing kernel of analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png as a function analysis_b8f76b26eedaad489d53d5308746b95a4e882871.png by

analysis_27ad83f1a2b1820ecc784de0228822b8a2707a78.png

From this definition it is easy to see that analysis_b8f76b26eedaad489d53d5308746b95a4e882871.png is both symmetric and positive definite, i.e.

analysis_755638a3595a7a1ead0e4e9612018e079d29c33b.png

for any analysis_84feeb83da8919c3f995550cdfb7e928ee8b390d.png, analysis_b89ede5a820dc8f099041b4d4f0ecdd4868687d7.png and some analysis_27633ac8c8aa0ae9d092b03465d6922c89a928d0.png.

  • RKHS in statistcal learning theory

    The representer theorem states that every function in an RKHS that minimizes an empirical risk function can be written as a linear combination of the kernel function evaluated at the training points.

    Let analysis_8c8d4992aa385c22d79f66db8edced75a661b728.png be a nonempty set and analysis_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png a postive-definite real-valued kernel on analysis_05c94a538dd234087167d26f9149a0dfacddcc9f.png with corresponding RKHS analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png.

    Given a training sample analysis_3de9cf03249406bdf6998f42330c359b742031d5.png, a strictly monotonically increasing real-valued fuction analysis_90c95329d50a07604a3ff41711c83187659ddbe7.png, and a arbitrary emipirical risk function analysis_4d123d60c7a0847580e4267a94c9bcb317466ce9.png, then for any analysis_f5ed078e87ca1fad56f1dec61314691e914f03c0.png satisfying

    analysis_adada7e83fa7569e066fc33a3ca0263269738988.png

    analysis_498a0868024087d893a089b85f7e4b73c64b5be7.png admits a representation of the form

    analysis_87db77eb5bfdb7fb8f5a71bcad299a3aee84a4c9.png

    where analysis_e4347a3a7874f372e87a8b16d8090a957a914720.png for all analysis_1ca89de6f655aa2165fe8a882b5603238943bb1f.png and, as stated before, analysis_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png is the kernel on the RKHS analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png.

    Let

    analysis_71fbbf2856448681481a02120db0f724f5ba4812.png

    (so that analysis_15f0e266fcbc4749f098d85d3b56d67904c03494.png is itself a map analysis_4447555113f2332b98f0bbfed30cbe60343d586e.png)

    Since analysis_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png is a reproducing kernel, then

    analysis_7c70c68bd72f300b77c09689e70eb2d04c6cbbcf.png

    where analysis_95e8e97f74983af288b98e83cca4b981427f69bf.png is the inner product on analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png.

    Given any analysis_527dc82cdc359f481ff1e436d9c8306e09011b62.png, one can use the orthogonal projection to decompose any analysis_7494375cd141e688509a55711b7f8388c5052bea.png into a sum of two functions, one lying in the analysis_9f77bb9ffdea2ed2a8d946d9005122531af1ba8e.png and the other lying in the orthogonal complement:

    analysis_014bda795dfa0fe4ef87cc28b22956fe904f0048.png

    where analysis_be5d877070745f63b78f73f12adcaab254e218a0.png for all analysis_f78e53d727e82c8e43205bb55f2368f6dc0affa3.png.

    The above orthogonal decomposition and the reproducing property of analysis_078b85cd3478400338e3a1ee425c2a468644be7e.png show that applying analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png to any training point analysis_4b07a6bdb6034c64c93f1dbc7477c11d15061d8a.png produces

    analysis_8727787712ab8ce3b626b9ffc42da23cd71a81ef.png

    which we observe is independent of analysis_d198c0d06b0da5525f9a966222601066509ac01d.png. Consequently, the value of the empirical risk analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png in defined in the representer theorem above is likewise independent of analysis_d198c0d06b0da5525f9a966222601066509ac01d.png.

    For the second term (the regularization term), since analysis_d198c0d06b0da5525f9a966222601066509ac01d.png is orthogonal to summand-term and analysis_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png is strictly monotonic, we have

    analysis_a9f61f3383538766411eea2e7d89255400aeda79.png

    Therefore setting analysis_c2965754872d5f83d04a502439d473f6a58b6878.png does not affect the first term of the empirical risk minimization, while it strictly decreasing the second term.

    Consequently, any minimizer analysis_498a0868024087d893a089b85f7e4b73c64b5be7.png of the empirical risk must have analysis_c2965754872d5f83d04a502439d473f6a58b6878.png, i.e., it must be of the form

    analysis_9d2e1b05fe8d903f6f7b97dd7270466c80c6ba71.png

    which is the desired result.

    This dramatically simplifies the regularized empirical risk minimization problem. Usually the search domain analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png for the minimization function will be an infinte-dimensional subspace of analysis_16ab4f50316bd4a289bdaafb72953dc027f482ec.png (square-integrable functions).

    But, by the representer theorem, we know that the representation of analysis_1e808437235fe06e358fc37a561bfc8a9551e448.png reduces the original (infinite-dimensional) minimization problem to a search for the optimal n-dimensional vector of coefficients analysis_9ca97ecfb81e55e2880cf9f32c25d932579b2948.png for the kernel for each data-point.

Theorems

Let analysis_221196b7b9b22212661a4c43996786b141f2aafc.png, then

analysis_58a2e702a0cd2cf9a06c38c8c67eeb7184ad754e.png

Let analysis_221196b7b9b22212661a4c43996786b141f2aafc.png. Consider the following polynomial

analysis_fcdd1725793099c3aeeb4bd2fdb080cb7419ab4f.png

where we've used the fact that analysis_c806fbd3e80a3afbb0eea4941b043760bec3bf16.png for analysis_7b1e0635ba9284c60652a4d373289297ff835303.png.

Since it's nonnegative, it has at most one real root for analysis_7a84c9a383f9772338016d101ccc096be06af784.png, hence its discrimant is less than or equal to zero. That is,

analysis_45b7ee412704322500988a34d8889fcbcc816f87.png

Hence,

analysis_613ad175d381115ca4e508e60c78615944a8c829.png

as claimed.

Let analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a metric space.

  1. A sequence in analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png can have at most one limit.
  2. If analysis_6a39f7b48d3fecf222e8e527ab907f1b3940b2f0.png converges to analysis_3778a663528a3e4db5ce811ed0c6cd9f5831579f.png and analysis_71ef8282c5e1bf94fffe20d1765d4241df4ee16b.png is any subsequence of analysis_3e0d749f8b54de075cf4b60155d416c2c52e85fe.png, then analysis_853a55ff44e71edc4f8dc6a88dca46223fd495ec.png converges to analysis_3778a663528a3e4db5ce811ed0c6cd9f5831579f.png as analysis_2703c2b1ae53a3e5be064c15b4bb53f564ae9ac4.png
  3. Every convergence sequence in analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is bounded
  4. Every convergence sequence in analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is Cauchy

Let analysis_6a39f7b48d3fecf222e8e527ab907f1b3940b2f0.png. Then analysis_8bfda3e1cf8db97f4eadc64747e2f64865dba7c8.png as analysis_2531823f4e49c847dd9a34d97de0ea57802b96fa.png if and only if

analysis_05027ae322a15055066af401170ba4e31e6459fe.png

Let analysis_ed7f570bd5e8cada8ea7e894b3dfc90eebb1847a.png. Then analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png is closed if and only if the limit of every convergent sequence analysis_48b41ab392fa66783f601bacdb6ee49f92bd9ca5.png satisfies

analysis_7b915e90d78f95329e9e0de0d93a1f48c2e64c07.png

A set is open iff it equals its interior ; a set is closed iff it equals its closure .

Let analysis_f7eec20146ea61ba7ca25392fa7ea5297cb502ce.png be a metric space and let analysis_04dbc02b1b2d96a53221dfd08c2e2d91d8a79104.png. Then, analysis_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png is dense if and only if analysis_2ae25c274da6f14d89f7719e5cabeb8211608836.png.

Any compact set must be closed and bounded.

The converse is not necessarily true (Heine-Borel Theorem addresses when this is true).

  • Bounded:

    analysis_3a31247bdcff52cd0254f74eab4d4072366fee5a.png

    analysis_1641d18cc980f8db14cdff95d7417a8526eef446.png is compact implies that analysis_be7d54c1b6e8fb2624edaec257275eec4493aa20.png s.t.

    analysis_fa6dc322ec5d7def1e3b4280e7c019446d09bce4.png

    where analysis_f15a1c621dd7843ab88ed63928071bf5fd54c2d9.png.

Let analysis_f7eec20146ea61ba7ca25392fa7ea5297cb502ce.png be a separable metric space which satisfies the Bolzano-Weierstrass Property and analysis_fbfd580f3ceb3e35a36797aa3dcf7cf005eeab89.png. Then analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png is compact if and only if it is closed and bounded.

Observe that

analysis_d67b8d600b141b903dbcd3b78ca871933a870621.png

is closed and bounded.

analysis_bb6264b41e6cbe7805162f414235bea08a0bbf9c.png is compact if and only if analysis_9e2f0b75198f1e75ddf417cd3a737985466d6bcf.png.

Suppose analysis_b965e0865a48d1d5cece6150491c1e1209ac5501.png. Idea is cto construct analysis_468bae3dd492e4f0b42e874723e3f35adf4e2c1c.png of unit vectors s.t.

analysis_d77530816d7e30291ae3b19cb6db6fff45a39060.png

Example: continuous functions

analysis_ed0d29f45cb69c338d019f5dd93a85c7383b6de9.png is a Banach space.

What are the compact sets analysis_fef374e0539ad221966bd252950c40744d99bc39.png?:

  • analysis_1641d18cc980f8db14cdff95d7417a8526eef446.png is compact if and only if analysis_1641d18cc980f8db14cdff95d7417a8526eef446.png is closed, bounded AND "something something" (what is it?)

Continuity and limits of functions

Let analysis_8fd511771ed24948876570c0e0ef76a099b637c2.png where analysis_f7eec20146ea61ba7ca25392fa7ea5297cb502ce.png and analysis_e538746b41a96f06d75c3146ece37c64df501acc.png are metric spaces.

Then analysis_6ebe84db719812f9f79fc46b87983b72143102ce.png if and only if for every analysis_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png we have

analysis_a3d4db3c83efecf7bead5b0755d445c4ca0a7ddb.png

Or equivalently, if analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is continuous on analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png,

analysis_bdd194ae1c8543f97733bade77238babf8378cef.png

Connected sets

Let analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a metric space.

  1. A pair of nonempty open sets analysis_57901127a6e46607502257d1b65f3be6ac59eb59.png is said to separate analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png if and only if analysis_2dab4a3ff817d592c8ef045e5466dfcab139fc9a.png and analysis_e0a09771728b4336e50bc88c6938e9751b53205a.png
  2. analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is said to be connected if and only if analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png cannot be separated by any pair of open sets analysis_29990d81cfd882592664e66495eef6da4c563fc7.png

Loosely speaking, a connected space cannot be broken into smaller, nonempty, open pieces which do not share any common points.

Let analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a metric space, and analysis_57901127a6e46607502257d1b65f3be6ac59eb59.png.

analysis_29990d81cfd882592664e66495eef6da4c563fc7.png are said to separate analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png if and only if:

  • analysis_845006b2da3ed1ebcefaac4d5fb3ea80cfc9459c.png (non-empty)
  • analysis_9e7cc33f04cb5c6a6cc712c0f87db351b6eabb92.png
  • analysis_e0a09771728b4336e50bc88c6938e9751b53205a.png

analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is connected if and only if it cannot be separated by any analysis_29990d81cfd882592664e66495eef6da4c563fc7.png.

A subset analysis_8bb977edb6466538b1ce951d2a306ca45fbdca10.png is connected if and only if analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png is an interval

A subset analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png of a metric space analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is path-connected if for every analysis_3e0475d40755ce5aa687a1dac0edc8dbe1ac336a.png there is a continuous function (path) analysis_b7ba1671de9798cc865ab7da43efda77b5fcac87.png such that

analysis_faacd3077d4294a8ee3672f9b3a11154d80097b7.png

Let analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a metric space, and analysis_797ea30ce809485708d5549bfdb870cab90ac7f0.png.

If analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png is path-connected , then analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png is connected.

Stone-Weierstrass Theorem

Notation

  • analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is a metric space
  • analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png denotes a algebra in analysis_f34c77864e3655ef36564478397e4dcd7aa15b30.png
  • analysis_c5e294383ca89a500db5fec4733e514c8b940d64.png
  • analysis_36b63b6ba52d44402cba8745131b1cc248fe88f1.png

Goal

The goal of this section is to answer the following question:

Can one use polynomials to approximate continuous functions on an interval analysis_b2fdf0617d736136c4564c3795fed8545c0717bf.png?

Stuff

Let analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a metric space.

A set analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is a said to be a (real function) algebra in analysis_f34c77864e3655ef36564478397e4dcd7aa15b30.png if and only if

  • analysis_e79fe718008ff0def4d3f31b38f72ec46407808c.png
  • If analysis_02faa4c8d8456c2ca51ee289769037e9724b89f0.png, then analysis_7462dd536b01138161b93884a77eaecae30da461.png and analysis_0bf126e4d8e11508a87efffc8c1bb3f94b9d72ee.png both belong to analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png
  • If analysis_8790acb079faad55d40b8b812d34883e82ef5d53.png and analysis_8e1920707fd0ecf563db5e5cf42dac8c316a64e9.png, then analysis_315be40b7402bb46307d62d711ccbe1414e4823b.png

A subset analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png of analysis_f34c77864e3655ef36564478397e4dcd7aa15b30.png is said to be (uniformly) closed if and only if for each sequence analysis_1a2de253563a847102713b3d8889bb1b3b071e3e.png that satisfies analysis_8f95996a2ee11d90669213b84aacbf5da13c2cdb.png as analysis_2531823f4e49c847dd9a34d97de0ea57802b96fa.png, the limit function analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png belongs to analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png.

A subset analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png of analysis_f34c77864e3655ef36564478397e4dcd7aa15b30.png is said to be uniformly dense in analysis_f34c77864e3655ef36564478397e4dcd7aa15b30.png if and only if given analysis_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png and analysis_5e9724100dbac385cdd3dfdbda759314035b7931.png there is a function analysis_2e6b391fb83e73b4e323b264e957fd058a71443c.png such that analysis_5c60a8108387c7ae03bacbd70eec47891a3c9cc4.png.

A subset analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png of analysis_f34c77864e3655ef36564478397e4dcd7aa15b30.png separates points of analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png if and only if given analysis_f7326e0f4b870a289df21daca43bb60c2343c016.png with analysis_d16f4fe600dfe23f9c4c2594f9c1043341f6ea1a.png

analysis_8b8279e85e37c7217a49ec8705e922d172f56f46.png

Stone-Weierstrass Theorem

Suppose that analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is a compact metric space.

If analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is an algebra in analysis_f34c77864e3655ef36564478397e4dcd7aa15b30.png that separates points of analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and contains the constant functions, then analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is uniformly dense in analysis_f34c77864e3655ef36564478397e4dcd7aa15b30.png.

This is HUGE. It basically says that on any compact metric space, we can approximate any continuous function arbitrarily well using only the constants functions and some functions which separates points in the metric space!

You know what space satisfies this? Space of all polynomials!

Q & A

DONE Alternative definition of compact; consequences?

What's the difference between the definition of compactness and the following:

A set analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is compact if for every subset analysis_ed7f570bd5e8cada8ea7e894b3dfc90eebb1847a.png there exists a finite covering analysis_040fb6dc800d9bfdad95f2d7e7583208d70225d5.png.

Is there any difference; and if so, what are the consequences?

Answer

Yes, there is a difference.

In our new definition we're only saying that the EXISTS some finite covering, while the proper definition is sort of saying that all coverings of analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png does in fact contain a finite covering, a sort of "the lowest common denominator covering has to be finite, and each of these coverings do in fact have this".

Fixed Point Theory

Differential Equations

Notation

Stuff

In this section we're considering the large class of ODEs of the form

analysis_d940e9d8b3346d0763159d7476f8d4be944c7b08.png

there will exists a unique solution analysis_162e83f551cdf0f5fee1a4751694f96acab63e49.png for analysis_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png sufficiently small.

To ensure the existence of a unique solution, we need to consider the following:

Suppose analysis_84343f833f55f42f91f522c91fe745b681e7d7e5.png and analysis_e18ae45a6495dd8fad904def37527a60bce28774.png. Also suppose analysis_a1805c342197ad3dbf7cfae004af60962120c37f.png and analysis_9504567bbe1f47c78a3ad4721e92e8009fa9bb09.png and

analysis_4905527a7e0c8b691fad89917cf14f647cd91431.png

is continuous.

Further, suppose that for all analysis_e30c596451c4e9ce09e0a399ab481c35eceba097.png and analysis_69bbf1b4400de3b26fa1921702563372e8bf689c.png there exists analysis_4174cb28db875db895768f049bab0e4942ebfd79.png such that

analysis_73918f5afd4963edc8ccf92ca84c2f7d4fc8e6c8.png

Due to Mean Value Theorem, if analysis_ce05d57b88ab29cfbc42446f6fb7b4054f6d9830.png exists and is continuous on analysis_a03a87866358cd77285905b563defe21989df62a.png then the above is satisfied.

Suppose analysis_b2db59aeb94cbc1050079892ff07b21b493513b7.png satisfies a Lipschitz condition as above. Then there exists an analysis_9b29b54e0b40c77a57b81caebd04f9c429fd621f.png such that the ODE

analysis_75c8a4f18718bed917bcd06ce09fede716f173ea.png

has a unique solution analysis_162e83f551cdf0f5fee1a4751694f96acab63e49.png for analysis_d4f8dee97c78867d2e1b1c41ce1f64403ce15527.png.

Exercises

From the notes

A contraction is continuous

A contraction analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is continuous.

Let analysis_f7eec20146ea61ba7ca25392fa7ea5297cb502ce.png, and let analysis_287bf5beddadd0da8587c5ba38d70f9fb077ae74.png be a contraction. Now, for the sake of contradiction, suppose analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is discontinuous at some point analysis_f4bda5a2a8228d16ebc8beabfd375043e905ee99.png.

Due to analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png being a contraction, then for any two points analysis_c7b83be083e2271b8705355bd245568af193a4e7.png we have

analysis_83150598c0fea82bb54040dc9e55f1024eba6ca4.png

where analysis_250173332a3b66ab3def45624080ed0d6e35fdcd.png. Since analysis_7a84c9a383f9772338016d101ccc096be06af784.png is arbitrary we can let analysis_7a84c9a383f9772338016d101ccc096be06af784.png be such that

analysis_0570fa9c7f679ad4cd2063a92b3d0d42c5bc91c2.png

Clearly,

analysis_0752dee7a53d67dd760e60084d9541e8c8c12334.png

Thus,

analysis_6c94c254b4b3c0c246d35ef0ab2661137c90ce78.png

Which implies,

analysis_3cb11bb0966402862a92ee74a3f3935292c76cd7.png

But this is only true if and only if analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is continuous; hence we have our contradiction.

TODO Exercise 1

Suppose analysis_287bf5beddadd0da8587c5ba38d70f9fb077ae74.png is a contraction mapping.

If

analysis_1ce0b5011055ddd1442261a02768563832c4a223.png

then any fixed point will be unique, whether or not analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is complete.

Further, show that if analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is not complete, then a fixed-point does not necessarily exist.

Let analysis_f7eec20146ea61ba7ca25392fa7ea5297cb502ce.png be a metric space.

For the first part of our claim, suppose the mapping analysis_287bf5beddadd0da8587c5ba38d70f9fb077ae74.png satisfies

analysis_3c66aa3e3ace76c02b566b786c23e53dc8633552.png

Then clearly analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is a contraction, since we can always choose analysis_250173332a3b66ab3def45624080ed0d6e35fdcd.png such that

analysis_dbb1c3dd71615142ee57898076263670bab31de7.png

due to the interval analysis_f093879d059361db2b2e498511abe868e7133db0.png being dense in analysis_5cb819dbdaa11557a460fe04a5ef95eede596daa.png.

Now, for the sake of contradiction suppose there exists two different fixed-points, analysis_3a8741bd160460254f30ece58103bd812d8456a7.png. Then, from the property above of analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png, we have

analysis_4611655938ca0c01bcf532f2d1cd13b2b78fe063.png

But, since analysis_220a09b1047609e2681fd330b426a5d9ee3f5c2b.png and analysis_fde70544aef3ef753dcd6a01e9e0e23ee9c28921.png are fixed-points, we have

analysis_c5cab7ed8c55a61b3c6d71104f719286897bdf1e.png

Hence, the above inequality implies

analysis_b88087b51a635b2860ad6daf91f6125510743470.png

Which clearly is a contraction, hence if there exists a fixed-point of analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png, then that is a unique fixed-point.

Now, for the second part of the claim,

PROBABLY EXIST SOME COUNTER-EXAMPLE THAT I CAN'T THINK OF. SOME analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png SUCH THAT THIS IS NOT THE CASE.

TODO Exercise 2

Let analysis_f7eec20146ea61ba7ca25392fa7ea5297cb502ce.png be a metric space which is not complete. Then there exists contractions with no fixed point.

Probably some counter-example I can't think of.

TODO Exercise 3
  • Note taken on [2017-11-18 Sat 15:59]
    Regarding the previous note, could we not have

    analysis_2c5f44fcfd704ae502f2c84cecaf545c3604a169.png

    which for even analysis_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png we would still have analysis_e0ca8447b8483477eed7c3540ddf6f24b13d5001.png be a contraction mapping, but for odd analysis_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png it would not be! Therefore I believe it's reasonable to assume that they mean for any analysis_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png.

  • Note taken on [2017-11-18 Sat 15:50]
    But analysis_f222ecc80cab9fa1774296c6eed6f5f3303b5561.png alone does not imply that analysis_b078f6bf935e2e55277c7e9d9fa01c5ac9537d7c.png since analysis_2df2cc9e7b226488f67d509eda18f07fec3e0a63.png, could be analysis_64d05c44c78da0a366853e18d698b5482d24c3dd.png, hence there needs to be something else which ensures this implication. That is, if the claim is supposed to be true for any analysis_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png, then yeah, this implication would definitively hold, but I'm not sure that is what they mean
  • Note taken on [2017-11-18 Sat 15:50]
    analysis_e0ca8447b8483477eed7c3540ddf6f24b13d5001.png being a contraction in a complete metric space => analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is a contraction in a complete metric space => analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png has a unique fixed point in this space: NOPE! We can have fixed points without the function analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png being a contraction, also analysis_e0ca8447b8483477eed7c3540ddf6f24b13d5001.png being a contraction does not in fact imply that analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is a contraction! See Exercise 4 for a counter-example.

Let analysis_f7eec20146ea61ba7ca25392fa7ea5297cb502ce.png be a complete metric space, and suppose analysis_287bf5beddadd0da8587c5ba38d70f9fb077ae74.png is such that

analysis_df3b4e142d082da4e9735ab6c6c38934f64eef56.png

is a contraction. Then analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png has a unique fixed point.

As proven in the first part of Exercise 1 we know that any contraction analysis_5631e2b393b07dcc9dbd19845c1ddb85cca1cb3e.png has a unique fixed point when analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is a complete metric space. Thus, we know that there exists some analysis_fe3d8b42293a7c7119494c5a50c513c0191b2d63.png such that

analysis_e4d8f031cf1faae5aafae581178f2b3b53a190e5.png

If we assume the claim is supposed to hold for any analysis_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png, and not a specific arbitrary analysis_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png, then the above implies

analysis_566e09623db67fff52b6ad76f74acb52d0f660ec.png

Further, if analysis_fe3d8b42293a7c7119494c5a50c513c0191b2d63.png was not unique, then

analysis_b2d667d86595f38eb07d48afc23ceec24dd9dd49.png

for some analysis_ac4c99c707d333b3959536c7f82a8df7d4af56a6.png, but this implies that analysis_e0ca8447b8483477eed7c3540ddf6f24b13d5001.png has another fixed point, which we know cannot be.

Hence, if analysis_e0ca8447b8483477eed7c3540ddf6f24b13d5001.png is a contraction, then analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png has a unique fixed point.

TODO Exercise 4
  • Note taken on [2017-11-18 Sat 17:29]
    Maaaybe you can build some argument by creating two linear functions analysis_62dfc834e046febca42e35b21e939f37b7e8817d.png and analysis_d5cecca75d6f76ab24a6b55f8ebdc1025cd242cb.png which intersect analysis_dbf24f7f5c65497bfe1c7c292726ecd334103113.png at a single point analysis_b2de3705f476de9dc306745a7b1dac8d4d3d2c48.png, and such that

    analysis_00a66dc092ee80740ce6242fa23e15d017310f76.png

    Basically, consider two functions for which we can easily compute the effect of analysis_9a1fa803b4f6996da4aa10b9eca1ce79b39f14a0.png for two points analysis_7a84c9a383f9772338016d101ccc096be06af784.png and analysis_eb83d466c7d035356e9f39998f357cee73da1e26.png on the distance between them analysis_6ad739a9f12614e0f9b06eadca62605db4c31d2b.png which we can clearly tell has the property that analysis_f0390fe72508b267e35cd4d2fe3e99ddf99a7994.png, buuut I'm not going to bother spending time on this now.

  • Note taken on [2017-11-18 Sat 17:08]
    But, if we can show that

    analysis_9f2e6cd3c54e9a17be60a9fe90069516fd7e7f92.png

    and then

    analysis_3214545c3ec546bf4a9e4cb4b8ebb9b5456f472f.png

    we're good!

  • Note taken on [2017-11-18 Sat 16:59]
    I was wondering if we can use some function analysis_93f821d500524ed5add6dd1440905bde651a16b6.png on the interval analysis_86ae464b52a7f6288d1c4cf40044cc951e41fe11.png which is defined such that

    analysis_c4cb3a4d26d18e29d643416a9146a97371ec9c9a.png

    which also has the property that analysis_0169b180a00ae7f5467af9644152b89e7a412bec.png for analysis_86ae464b52a7f6288d1c4cf40044cc951e41fe11.png, and then we potentially compute the distance of the difference between these functions to obtain some upper-bound on analysis_dbf24f7f5c65497bfe1c7c292726ecd334103113.png which related to analysis_0a07962151ff3d9768d4a7693d7575a1c090e0f6.png.

    analysis_30825b4560354b2e3eecc5fd92b67874bcf038eb.png

    Doesn't seem to work very well though

  • Note taken on [2017-11-18 Sat 16:34]
    Technically we only need to prove that analysis_52d7cc3f2ef703e7dcc9b5d246ed21c14e2ea553.png is a contraction mapping on the open interval analysis_6cddbb9476d4e39baec5eceb188a3d5a03d04e3b.png, since for analysis_7a84c9a383f9772338016d101ccc096be06af784.png not in this interval, analysis_8a0a367b8a6d965da5692a08037dc067ceadaade.png will lie in the interval specified before.

analysis_1426036d3b737fe53400029f9ad012b29cd126b2.png is not a contraction, but analysis_b15c37eb9b3c09c452100bbf477284f62721e7a8.png is a contraction.

Further, this implies that there is a unique solution in analysis_492d525117d0dcc93d066c8759f46b98cf9980ca.png to the equation

analysis_0c451336905179e1e89404137a117ab05c7cd9b9.png

Clearly analysis_b784d34dc6deff8e5b147c5494d95a3ddad83e99.png is not a contraction, since

analysis_8a8f11c538ccb047e3fc7d499dc19b6b3b056214.png

which implies

analysis_7b2ff9edba716ac3c1b12a8e037988d19d799bbf.png

and

analysis_c206face3c7e16ae8ab3e44ae7e92421851e4669.png

Hence,

analysis_12bd359b18b813e2c4c8039c1c27ed181ea62448.png

NOW PROVE THAT analysis_d713921132cee858cac45bb3a883fbb5f3d8a93c.png IS A CONTRACTION MAPPING YAH FOOL

Fourier Series

Notation

Definition

Let analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png be integrable on analysis_3bba7d56d1b43b65d087b08707b278935c882c1c.png.

The Fourier coefficients of analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png are numbers

analysis_46401906f2742763b7bb097015a6759768fd368a.png

and

analysis_f679d90162bb7363a3966c706431f3da5a9e801a.png

Let analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png be integrable on analysis_3bba7d56d1b43b65d087b08707b278935c882c1c.png and let analysis_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png be a nonnegative integer.

We define the Fourier series of analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png as the trigonometric series

analysis_0abb8c52a536006160afc290505c0df510bd2eff.png

and the partial sum of analysis_6492518b997df628c31e659e47ebecc9647417ab.png of order analysis_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png to be the trigonometric polynomial defined

analysis_c27ed5114acc689ff4eface43ad0aec5a4c81702.png

Kernels

Dirichlet kernel

Let analysis_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png be nonnegative integer.

The Dirichlet kernel of order analysis_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png is the function defined

analysis_f39ae82f979f41c13d642f6439de49f195cc4c1a.png

with the special case of analysis_3a714baa6d6519ce1b1e393ef3cc8e2dcb5da27b.png.

It turns out it can also be written as

analysis_da69e5ebe5bda9d422b7642f42a45ef2a0fee4ef.png

analysis_8bb88566924b55cb6e90edd8d2e5b0ae3f90d797.png

for all analysis_3fe5c68b8d5d0b34932327019a59621d392c217a.png and analysis_7c894fec43a3124cb3587458b79302e97436674a.png.

That is, we can write the N-th order Fourier partial sum as a convolution between analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png and the Dirichlet kernel.

analysis_3d8143e67e2cf40cac5a525a15a8f1ad9a3bd3d8.png

where we've brought the intergrals together and used the trigonometric identity

analysis_195f3e0e8d00d5933fad2258fed262a0ac7ae93c.png

Finally remembering that

analysis_a4fadeb4d497a714cbe2d614e7b802e5d3838bdf.png

We see that the above expression is simply

analysis_0b3a23f175a8df5981dce512cf5afb93a655f002.png

as claimed.

Suppose that analysis_7c9e4f5a73e24e9c920b6f8af480d6a20bd68db7.png is integrable and that analysis_d5bfce3cf999dfe9c8f51b302cf828f84b901a52.png uniformly on analysis_3bba7d56d1b43b65d087b08707b278935c882c1c.png. Then,

analysis_af0d2f780a8012add6e3cf06eca48f282423f3be.png

as analysis_d42970d8ef75b6eb77b66c571a7ec127029eaef6.png uniformly in analysis_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png.

We know that analysis_abab5de33eecb551cc83e15bca757c0e3d5c8601.png such that

analysis_76b95d441d1310a75550bb6d5e873aa1457e1b7c.png

Then we have,

analysis_a27f0f3146cb5d28944af73256c3e787f1e4caef.png

I.e. we can make the difference between the coefficients as small as we'd like, hence

analysis_19f2db9516b90d31901d09b222f199ad7cfef5df.png

The very same argument holds for analysis_7bb0bc6108dfa42aab51cbb68fc4765623b52316.png.

Fejér kernel

The Fejér kernel of order analysis_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png is the function defined

analysis_ad644b979a3388e4b3c3adafc4bffbf30b5528e0.png

with the special case of analysis_3a714baa6d6519ce1b1e393ef3cc8e2dcb5da27b.png.

Functional Analysis

Notation

  • analysis_e905228e61b3405a1c5b39c0d05f6d51dace2ba9.png is the fields we'll be using
  • analysis_a72359b7902a685d57bd5c8dd1d8194f2d1e97df.png
  • "Small Lp" space:

    analysis_5d732b385aff002888f344223c6e5b23e3109d61.png

  • Let analysis_15d54f7fb75d3425948028f3d67c5d54e2085b29.png be a measure space and analysis_1f09ba8dec07810eeac884155e764222427bb4dd.png, then

    analysis_9307497492f677df57c16b31ecad293d9de9f14b.png

    denotes the set of all measurable functions analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png on analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png such that analysis_63ed33091210d0311a3e3f84a6cef905d1d9838a.png and the values of analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png are real numbers, except possibly on a set of measure analysis_96f53e8f2667720f54bd85623f46cbf545733989.png.

  • Lp space:

    analysis_544e07a9ef343812dc02600870ecdae4e4fec194.png

    where

    analysis_c412d66b54850cf82977e6e3d08e4187309da5a1.png

    Or more accurately,

    analysis_33578f298e71329a2cd70dd6203d2575daab807e.png

    where analysis_278b27a2fe3579b74161c096e5a172a21f7aa4f6.png denotes the equivalence classes of the equivalence relation

    analysis_5349ce82d7480846844afe88955c1241a585e0ab.png

  • NLS means normed linear spaces

Theorems

Inequalities

Let analysis_8fce9cff4b234be8762736ce02b1233f91936096.png such that analysis_a29a43ab5b4ed12e0741feef78b3b3d8c89ec4ed.png. Then

analysis_23b7f3a43426985d4b36c0a1a9c270f207b284d6.png

For any two sequences analysis_c481b027534bffa013e9e109fabee66e9994b517.png and analysis_975f1a851eb6acafe5876ac483d60d8dcd254803.png of nonnegative numbers, we have

analysis_d788523fef2f4ba4c05c915430a7b7dfbf0a663b.png

for any analysis_a8b2ffc14dfe88d9926c73005742162e201665bd.png where analysis_f63749027365e29025a5cb867262c465d2de65bb.png is the conjugate exponent of analysis_7225b076f6e6326f1636b11d1aad8de58bcc4761.png, i.e.

analysis_4098b61e0d8198243a6f363bb904d376c5dd6416.png

Observe that this aslo includes analysis_1e21c72ab9d09961058aeb77592e6cb2fb103bae.png and analysis_cd751b64eb13fe69817b49abe89f699403a3dafd.png!

For any two elements analysis_b1281cced21e35ecbb22f7cc633af85cd23f678a.png and analysis_d334811584d910f64d6b699580522e9e4edb3b3a.png of analysis_7c0c46e1815c22704247ac14b8b59a09d4788b24.png, we have

analysis_13620a1ee69880e734280f22c30c47dde65ae693.png

for any analysis_a8b2ffc14dfe88d9926c73005742162e201665bd.png.

Mercer's theorem

Let analysis_ce8a4ce3819875416474aa2dcf1372ccf5b78fbd.png be a symmetric continuous function, often called a kernel.

analysis_1641d18cc980f8db14cdff95d7417a8526eef446.png is said to be non-negative definite (or positive semi-definite) if and only if

analysis_4ecacfbb519598ddb50c6760b53bdd3d3d208703.png

for all fininte sequences of points analysis_09b35293fe0907cef00c439b793448e7e37b0926.png and all choices of real numbers analysis_35118d99a974a7a36d09cab3dd36c44ba1e593c2.png.

We associate with analysis_1641d18cc980f8db14cdff95d7417a8526eef446.png a linear operator analysis_20b144bf5d2c32b9f4833c62e4541eeefe3362c9.png by

analysis_b3bbaddf17cba9622a10a6293be4a7a06437878c.png

The theorem then states that there is an orthonormal basis analysis_127cb270e3d9d0f41354146d98652b6ee5ee8b95.png of analysis_f0c94b7d2b5f0f0c4476ee6445739f9887e06e35.png consisting for eigenfunctions of analysis_693ec4b7194b2efe9256b21a1f8a855c5e198eb2.png such that the corresponding sequence of eigenvalues analysis_db703d5499816e76f608b8777b0088756ba6da45.png is nonnegative.

The eigenfunctions corresponding to non-zero eigenvalues are continuous on analysis_48286279ea8482e9b623863252fcc0f2c6430f89.png and analysis_1641d18cc980f8db14cdff95d7417a8526eef446.png has the representation

analysis_6b54fc32b9ce488e504181e7a5a6e3f1de95d6f3.png

where the convergence is absolute and uniform.

There are alos more general versions of Mercer's thm which establishes the same result for measurable kernels, i.e. analysis_1117bfc670de39be88cd023892ec7fc810e8bfe6.png on any compact Hausdorff space analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

Banach spaces

A norm on a vector space analysis_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png over analysis_9360329fce6f3b05ed0cad8714426e714a6dcf1e.png (analysis_cf2cc88b588da2b47242417c006bb7a585b6076f.png or analysis_20a24d1ed449dcbb47090631b2f93ca1189c3c42.png) is a map

analysis_08e3b028503611b7c37ecf1c14ca5822854115ad.png

with the following properties:

  1. For all analysis_2057047b854476729e402d900a0dc2686aece479.png, analysis_6177bc72b45d35e96a49a87640333fccff50d29e.png, with equality if and only if analysis_024981e9e4d3820bc6b79e575eec833786942650.png
  2. For all analysis_2057047b854476729e402d900a0dc2686aece479.png and analysis_499f266e68be7ff16f8a9bf85f95a924080d865a.png, we have analysis_f26b1c07b032e708ed277339e36a5d78299f434b.png
  3. For all analysis_eb377bd413330e7d6a3e1d666614d114b6b5a3ac.png, we have

    analysis_db0ea2f0b52b5384d0bb1d2a7cd2178fdd58b130.png

If analysis_91e92998c43770595cfa6e0f4e74def49e43f939.png is a norm on analysis_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png, then we can define a metric analysis_f4e5566de1c75b7700b562b91a80d2308cc66d12.png on analysis_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png by setting analysis_79fb302542967cf12086e9371ff8966379aed49f.png.

A normed vector space is said to be a Banach space if it is complete wrt. the associated metric. A Banach space is said to be separable if it contains a countable dense subset.

Let analysis_a20bca6b2b1dca2278ae5b7fbdc04932be035b8f.png be a normed space and analysis_8f593a0a71693446eccc9f47248035f8cecbe561.png a Banach space.

Suppose analysis_f53cf2c3985c80987bd9707fd5a0b28594db886f.png is a dense subset of analysis_a20bca6b2b1dca2278ae5b7fbdc04932be035b8f.png and that analysis_c34c87c0298deef8f43273ceb2336a6e0dc2c0cb.png is a bounded linear operator.

Then there exists a unique bounded linear map analysis_802d78c0dac9cee5d236c639f19011694789fd97.png such that

analysis_c6c4dcac75c813042d1d3be77c362271b0d5be69.png

where analysis_fd837ff400200cb180ae2e82c3aa4c45fe99b037.png denotes the restriction of analysis_94b930184c99d7c6ab52f51c3deef85b348787f6.png to the subspace analysis_f53cf2c3985c80987bd9707fd5a0b28594db886f.png.

Furthermore, the norm of analysis_94b930184c99d7c6ab52f51c3deef85b348787f6.png equals the norm of analysis_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png.

Suppose that analysis_a20bca6b2b1dca2278ae5b7fbdc04932be035b8f.png is a Banach space and analysis_8f593a0a71693446eccc9f47248035f8cecbe561.png a normed vector space.

For any linear map analysis_e0b095c718fe8edcfa2d85aed6555cf654f9dab1.png, let analysis_ae7250df7fefbd0d10242eff0309aca69c932b6e.png denote the set of pairs analysis_3ccba1c962c5482c79e366b6761545336c307d47.png in analysis_a03e89521cb41870baf49128113cc4f4e5a43bd8.png such that analysis_fa129df5c62cf730d29892393fb4f37f9e35f4f4.png.

If the graph of analysis_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png is a closed subset of analysis_a03e89521cb41870baf49128113cc4f4e5a43bd8.png, then analysis_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png is bounded.

Examples

Equip analysis_b9cafadeffc79ac16c260278c4a3676fbb772e26.png with the norm

analysis_2a41b1ae56c4731da7925d5fc114ffbe835b22dd.png

Then,

analysis_3d34c30b5e3750531c27811d3d5312cd5d469f20.png

Metric space structure on NLS

Let analysis_433b1ab75dc7520f3bf952994f7828c3378c1cb8.png is a NLS of analysis_28ddede68ff90e4a1a646688039b4084c62a397e.png is a subspace.

If analysis_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png is open, then analysis_9855cffb8ead2133c08cbb1c9441b4a8adf4047f.png.

Clearly analysis_67cfa7bc4219fbc2e2a6a7d2d765849472c3578e.png, hence analysis_20b349404b167767fc901e18154c1ad42e54ccdd.png there exists

analysis_b24d3ff1b75beabf61ee61c30daffd87a1bc1174.png

Observe that analysis_72819678506dbc9232555f769e2c30c2478845af.png if and only if analysis_097f1d42112ed5c1e9d237fd9fa8ea70e1ec8252.png for some analysis_1f195de9aaaf5254222f55b0f111d2808775ac5d.png. analysis_85a48dfbdbafa5eb610a72628a4edeb4106be807.png by definition of a subspace, and reverse is analysis_21499886dc569490627f7066995f886597ce721c.png which implies that analysis_b1d329543a7f29b5d2cee929b2de2217a29ba867.png.

Cosnider analysis_fd4d33103d4ef9cada87f6e22ab3b456ed7e4911.png since analysis_e5762233835daf33218079dc95e5f43f75ca7312.png. Then,

analysis_4b692707dc6c93e33a5f7e04bad200bfbef62b6a.png

So basically, since a linear space is closed under scalar multiplication, any open subspace analysis_04dbc02b1b2d96a53221dfd08c2e2d91d8a79104.png must contain any scaling of analysis_40fb973cd997849a029e392c385d32d4b8c40196.png, hence it must contain the entire space.

Let analysis_433b1ab75dc7520f3bf952994f7828c3378c1cb8.png be a NLS with analysis_9e2f0b75198f1e75ddf417cd3a737985466d6bcf.png.

Then analysis_04dbc02b1b2d96a53221dfd08c2e2d91d8a79104.png is closed.

Completion of NLS

Notation
  • analysis_433b1ab75dc7520f3bf952994f7828c3378c1cb8.png denotes a NLS which is not necessarily complete
  • analysis_9d859cb50fa691bd916656e2bdcb223f268aafd4.png denotes the completion of analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png
  • Set of Cauchy sequences in analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png:

    analysis_ea49dac0c06efd997a410de85982fcd5d51a30ee.png

Stuff
  • Formal procedure to "fill holes" in an NLS, i.e. making non-complete NLS (NLS for which Cauchy sequences does not necessarily converge), well, complete!
  • Observe that analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is dense in the completion of analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png, i.e. analysis_d552c84426a409b27ac6e8153a2890888eaf95d2.png
  • Let analysis_ec0aa28c7ceb0ea01a15e40aa6b26ec8217037c2.png.

Let analysis_433b1ab75dc7520f3bf952994f7828c3378c1cb8.png be a Banach space, and let analysis_04dbc02b1b2d96a53221dfd08c2e2d91d8a79104.png be a subspace.

Then analysis_f0d6163f2cf3dde6a41bd7e151046bfc48e78a6d.png is a Banach space if and only if analysis_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png is closed.

analysis_f9a37dc730cf12c09ef495ac3f5c8057b90caf64.png Suppose analysis_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png is closed. Let analysis_c5b1cabad47eda4534e41f43882cd623d73ed2f1.png be Cauchy. Then analysis_e6cdf4866830b5a5e4b9d91b46225b629e9fc435.png is Cauchy in analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png. And since analysis_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png is closed, analysis_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png contains all its limit points, hence all convergent sequences in analysis_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png converges to a point in analysis_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png, i.e. analysis_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png is Banach.

analysis_6081d1ceb52e430c06c79ee9ca315c86854fbd32.png. Suppose analysis_f0d6163f2cf3dde6a41bd7e151046bfc48e78a6d.png is a Banach space. Let analysis_c5b1cabad47eda4534e41f43882cd623d73ed2f1.png s.t. analysis_97621c3cd88296433cd36d03df345c6772bc32c9.png for some analysis_4c91032750887d8dca229b906727af58e9268232.png. Note that analysis_e6cdf4866830b5a5e4b9d91b46225b629e9fc435.png is Cauchy, and since analysis_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png is Banach, there exists an analysis_7a075e23fa4c37d71d9b806c512c1354f7140ca3.png s.t. analysis_b4b5235fd9e2c63a2dc48de08952ec4e1b428fde.png as analysis_2531823f4e49c847dd9a34d97de0ea57802b96fa.png. Then

analysis_962733f9401742e872a1a7ae21a65640b288adb2.png

Hence, analysis_87f0bc54f5e9c00e78228b6e0e9a3b1822ab3fe7.png, and analysis_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png is closed.

Let analysis_d6bdf921e5aec0d473f6c8c2c79a8119f0861f9b.png and analysis_3f8966fc859a260224c388ea5c6f7dad6fd73813.png be two normal linear spaces.

  • A linear map analysis_b8f3bcc04a0c7cdefbcdd96b4640ad78f4fb15fe.png is said to be an isometry if analysis_3ca2928a09c1da1d0d3601d0b1995621bca9587d.png for all analysis_4c91032750887d8dca229b906727af58e9268232.png.
  • We say analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and analysis_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png are isometrically isomorphic if there exists an isometry analysis_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png from analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png to analysis_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png.
    • Note that analysis_8bd4cf796044dd401d2f62b0d999a11904bfcd62.png is automatically a surjective isometry.
  • The Banach space completion of analysis_d6bdf921e5aec0d473f6c8c2c79a8119f0861f9b.png is a pair, consisting of a Banach space analysis_3f8966fc859a260224c388ea5c6f7dad6fd73813.png and an isometry analysis_b8f3bcc04a0c7cdefbcdd96b4640ad78f4fb15fe.png s.t. analysis_c177341b91f875be1f5e661135098389013339c9.png is a dense subspace of analysis_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png.

Let analysis_54fdfba617de9943bea662fc0a6ad7afb09710ca.png and analysis_47405ae7f079e663f1ab6f73ba44fb6feaf0b4eb.png be two completions of analysis_d6bdf921e5aec0d473f6c8c2c79a8119f0861f9b.png.

Then analysis_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png and analysis_ca2bcc80309e331f71dd2c1e1f4788cbc048c242.png are isometrically isomorphic.

Let analysis_d6bdf921e5aec0d473f6c8c2c79a8119f0861f9b.png be an NLS.

Then there exists a unique a Banach space completion of analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

  • Let

    analysis_40204215067a8cd2a53cc02267028629b412c94b.png

  • Equivalence relation analysis_230220721c63703fdbe33a5ead0093180ae7e5c8.png on analysis_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png

    analysis_297fe25ec3fd5eb3761d3d2fd01f6de4f30e1db1.png

  • Let analysis_090d7f7a17501efdbff0aec311c2e2aaba4c6241.png, and define

    analysis_5455c0a23e2b8b6750c1b2a5a0951a3fffda8351.png

    • Observe that

      analysis_c66253cc58c1bd1cf3c51c4015407e82b99c54bd.png

      i.e. all sequences which converges to analysis_7a84c9a383f9772338016d101ccc096be06af784.png.

  • Equip analysis_811cc619d8e0a0f701f949c55a2960ab3aa75fb4.png a vector space structure, i.e. addition

    analysis_e69d8f90f4ffa040f6f76398ab927f889aeaebed.png

    and scalar multiplication

    analysis_ee351560ddc4cd4280e028a50ca77889ca7b1e06.png

  • Equip analysis_811cc619d8e0a0f701f949c55a2960ab3aa75fb4.png with a norm analysis_b173c52abed91d7712540a528e0f159dcfadcb37.png

    analysis_4933c800a287e6350d407356a72b94d741309295.png

    • Observe that

      analysis_3e7b5fa4c9c0678cc60e300e679f988c36baaf96.png

      By completion of the underlying field analysis_9360329fce6f3b05ed0cad8714426e714a6dcf1e.png, analysis_60da17281573fa01119d7954fb96e314ceba0517.png is Cauchy and thus converges. Hence the above norm is defined

    • Need to check that this is well-defined:

      analysis_ad512fa257614bfaec7bbc4eec8cf4176ef0d14a.png

  • analysis_8c484d358159efeb287dc48b79f4d56c403abfdc.png is a Banach space
  • Then analysis_592a724df1e49e1e190632175b118e40e95298fe.png is a linear map, then

    analysis_e0e733360d4d10ff077c25ea2ebddfc6c1880d8b.png

    Finally, one can show that analysis_c177341b91f875be1f5e661135098389013339c9.png is dense in analysis_811cc619d8e0a0f701f949c55a2960ab3aa75fb4.png, hence we have our Banach space completion.

Example

Let analysis_8b5fda39e29cad1c68d41776522087c7e4c0042d.png be a subspace.

analysis_82865b7ddd9f2628b05983338ea11b212ee202f1.png, i.e. analysis_1af46bea1bcf8a48c8d510ae629e07c7474f5efd.png is closed in analysis_e7ff2c42c06756a77419b77f72cca61efb0e56fc.png.

Let analysis_8a14112009d413418d3016a6179dc851c94f58ea.png, thus analysis_25f66d898eed7a0168565a0269d5926356bd7dd4.png. Then

analysis_9738a61a2fd0ba98ef0a2906be7063d402e77d36.png

for some analysis_fd115581effdd26b816cab0a6fe79b14b382a997.png.

We then need to show that analysis_eec913dc73095c789f66f06f017a8b87b8f2b2e8.png or analysis_48b04a89b7378db454a218715375e4d8da2b79b1.png as analysis_a4c8dc9d99a55084552339a4e055c35b9272ba80.png. or

analysis_99afe8ec9b51be1754e56589cd7d164b91c6ea4a.png

Given analysis_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png. Then analysis_54834ce815eaa25f6415da5e2a7697880a112f99.png s.t. analysis_4d25840b25fd6194c7ddf497f3096b753290c70a.png. But analysis_6f3d4d1f1d26df21375a546e50ff3cf3e473589e.png implies

analysis_ed001512b4ba8b376a7ee38a7720b82a85aa6cad.png

So for analysis_141614f1439ee04a54fbdf5f3e42608b8a0dff6d.png,

analysis_0862195e1f47687f96721d3869cf54de4343c690.png

So basically, show that a sequence of sequences analysis_dda305f36de44de22701d399d6d2f7cd8a2b6f58.png converge to some sequence analysis_eec913dc73095c789f66f06f017a8b87b8f2b2e8.png, thus analysis_1af46bea1bcf8a48c8d510ae629e07c7474f5efd.png contains all it's limit points, hence analysis_1af46bea1bcf8a48c8d510ae629e07c7474f5efd.png is closed.

Consider the NLS analysis_38466b84cd608c6a951ef61259d59b726b26f1c0.png for analysis_6cf56ebaec8a032a42b2a4f6d19f4c91ea8cf2a6.png.

Let

analysis_d07bda027803e865b30a74738aec8bebcd0173a1.png

Then analysis_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png is not closed, but analysis_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png is dense in analysis_f64c6c1f9f56923793bd78a927b47ea43daf66e5.png, i.e.

analysis_1b9a215b154cc1847fbd6edb36ec5833fdfd2b7b.png

Let

analysis_4a296e95797a3e3c914933ea20186279c45c1f8f.png

Then analysis_02e2a96f708bc20357a760656a40b89270b5d7ba.png, therefore analysis_0f4816c5358bb532518afb067c3528af2b7c3908.png. But analysis_0a066a06765b34c8e0c4776cefdfb77a34036bc3.png, but analysis_f6d396c8c9c6f4ccbffc13d883ac443a69d704e2.png, hence analysis_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png does not contain all of its limit points, i.e. is not closed, proving our first claim.

Now suppose that analysis_1fa8d4c033d2dd96584709fd15964d2600ee9380.png. We now want to show that analysis_63d9b0bd4c4cd9a578c2923dec80ead5c97bd5ac.png such that analysis_0f4816c5358bb532518afb067c3528af2b7c3908.png and analysis_86fff707df410a65631e9a7f612d32ddf7486c48.png.

TO THIS.

Equivalence of norms

Two norms analysis_da103b2c0a2d2d42eeec42393b51907f5c1badd7.png and analysis_bcd3d454eafddd86bea9e29756db432f17450fa7.png on analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png are said to be equivalent if there is a constant analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png s.t.

analysis_ee462a340e55542b7107748586ed2d40d2608d45.png

This does in fact define an equivalence relation on norms.

Hilbert spaces

If analysis_f3771cafedae120e5569c2845958c3e8409a9b92.png is a bounded linear functional, then there exists a unique analysis_13dc4cfcd652fc796f4174036b7702c21e046a88.png such that

analysis_fd7cfe272f615376806622ba2dc3f2b37aa694bc.png

Furthermore, the operator norm of analysis_42b688f7ebfaa0e3791010e7725767cadf2ad420.png as a linear functional is equal to the norm of analysis_4ef26ff1bd3f789ae16862cbfc47f13a48b0f7eb.png as an element of analysis_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png.

analysis_ba9aa955c3c566b4c3983fea7247055e19e01465.png and analysis_eb3bc451c199903da28ba45b6f29f0118096c5c5.png s.t. analysis_34f56a5f311192bce8975ef5afc0c9e7181a1272.png

  • Assume analysis_8cbcb18ca650893bf42d898c434c804bd1ae90c6.png and consider

    analysis_ed96d35a1625e3b35635a9753d7d4e208d082c74.png

  • analysis_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png is closed (analysis_31a25bbd38d2362b6bbf9ea118a2d106aad36453.png cont.) and subspace (analysis_31a25bbd38d2362b6bbf9ea118a2d106aad36453.png linear) of analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png. Then

    analysis_245e11fc7c72ac38e625cce69796e4797a4430a9.png

    since analysis_8cbcb18ca650893bf42d898c434c804bd1ae90c6.png.

  • But analysis_c0efb2a4d967dc62a157d25acec7c8562ba73fa3.png implies there exists nonzero analysis_e3e09f470fc7b6a27d7fbf250a40003a4abfe66d.png.
  • Take

    analysis_8cbee64ecb6d64d5dbd3e95d35b57fa8c6259283.png

    and check

    analysis_36ef97f12a77632e76b2bf3423373db0417396d7.png

Basis in Hilbert spaces

Orthogonal decomposition

If analysis_c600cd6cbc1d84ae21b495d794980130664c3dd5.png, then

analysis_865398c9cba006c14661da3e84306af39ed5f107.png

For any seminorm analysis_91e92998c43770595cfa6e0f4e74def49e43f939.png defined by a semi-inner product and any analysis_eb24e8bbad25acb50caa86f71c120365c2470d86.png,

analysis_8d52bb1ef8fd5083e7f8244e65d6b7a4ba282bd3.png

For any inner product space analysis_f854480bc0d9a8f205ce374efeb6d1488b2dc02e.png, and any Hilbert subspace analysis_25d1a2e2f2fc866febbe778231adefeb0ff587ed.png, and analysis_43f62a5f9474878c55a1c72b3ae3b6ae15d10906.png there is a unique representation

analysis_0452f6862d0758f7603fb4eefd488670f533e248.png

with analysis_27b6c3c32a739d4179fdf55c13b2ae01a0300ff5.png and analysis_f372a628d5df5218cc20b3eff226134c4c903668.png.

Idea:

  1. Prove existence of orthogonal decomposition
    • Proof by contradiction
    • Consider a lower-bound on the distance between analysis_0ec3e370519c30a01ee40393705503ee68249366.png for analysis_4100ea22c75ff38f4db49d8987335d2f16041673.png, and then show that this is violated if analysis_c0174eb759fe5b443f03936f62c91e13c30c643d.png, i.e. the "rest" of analysis_7a84c9a383f9772338016d101ccc096be06af784.png is not in analysis_c046e61e2a96a5402b0fad5d5fac395c90c98569.png.
  2. Prove uniqueness

Let

analysis_755ecad4f37ef4397c14153288c858321b1cdd2b.png

Then let analysis_8ae2a5fe96c8a6f60d9dd27f9c09f017fc6b1468.png such that analysis_fdf2235dc79498f17cf0cbea40ead6a6fb230941.png and analysis_3aa9fa59f8311379799235787847e958859340d2.png.

Then, by lemma:parallelogram-law we have

analysis_6afd8fe5ce8ad6d72d1961f258964a1a7fb55960.png

thus,

analysis_c9f3f54da69113a835a48e574f8037713ae3f454.png

By completeness of analysis_b2db59aeb94cbc1050079892ff07b21b493513b7.png, we know that analysis_d9ea49a420c745d6f294249d9b3e1f92e36cbc36.png. Further, observe that analysis_2e1a85caea0a1e7a6b6678fe5668ee0aba13c55e.png, thus by def. of analysis_cc6b5c35669c3b0901ef37cf78827ca69227d7b2.png,

analysis_c92efef6b45679ceae17a25b7a607bdbea057bfa.png

Thus, as analysis_cbf4d3a740a6baf8460cc5ac37fce15e71dcc170.png, we have analysis_c9f0f42fa29c15a32ae6a39840ca5a3eb9930327.png for some analysis_27b6c3c32a739d4179fdf55c13b2ae01a0300ff5.png.

Now let analysis_996f542c3951301b37447995fd302960d988135f.png. Then

analysis_7918338a252ff6aefc0fa60d0fccc0fbd59450c6.png

by continuity of analysis_84d9d552378b49d6862906448ec3cab2d0be627c.png and the fact that analysis_4347c3930333a84ab9ab609d2b66edb144ca48cc.png and analysis_c9f0f42fa29c15a32ae6a39840ca5a3eb9930327.png.

Further, suppose that analysis_c0174eb759fe5b443f03936f62c91e13c30c643d.png for some analysis_4100ea22c75ff38f4db49d8987335d2f16041673.png. Let

analysis_8e955cab7fbc600857df98c5712ee9c4f0607182.png

and analysis_69f835f52ca5b8a4d10991e388a61b407d13e6b0.png.

Then we observe that

analysis_b9c3018db16e5a48b475dff23fb8f17397b1ed83.png

Substituting back in our epxression for analysis_8b0e7a5ea4d56f5afdac3c2658ca87bd50b0ad6f.png:

analysis_edd29767cd34364ca068c84c892dfe0e726c4fa8.png

The last term is simply analysis_69c49267f691e34ea9ef992d59026b4bde55ec41.png, thus

analysis_0fe4569da5d41ed45d2c1baa9a2a9693bf0dcbd4.png

For sufficiently small analysis_d198c0d06b0da5525f9a966222601066509ac01d.png, the analysis_d198c0d06b0da5525f9a966222601066509ac01d.png term dominates which implies that

analysis_4be7908d556e5704ee2a509cdfa5aa0845e54346.png

for analysis_c75582d33dece273e09a75dc136a4d81ceaf35a7.png.

This clearly contradicts our definition of analysis_cc6b5c35669c3b0901ef37cf78827ca69227d7b2.png, hence we have the proof of decomposition existence by contradiction.

For uniqueness of the decomposition, we simply observe that if also analysis_4ba51a47ca062298ce5e89dc8e9cec5b24e65e14.png for some analysis_6e59bfc050b78487c3cabb2e5e57c898c9692a99.png and analysis_7b41ec5ac0d4e1487e05d065f7594ad56976af1d.png, then

analysis_64b9829045b26e85fafa31f4da40c681110ff1d0.png

thus

analysis_0a761672526d0df4572cd011e88c86dc3b7fcbaa.png

which implies analysis_034ac7f7844a625d3ce4c4bce6331c6a40937ab6.png and analysis_e5cbe5a5ddc500839cb60041d66a55a1f9a59fc7.png.

Many functions in analysis_c4008d3da4349418a466832c573efdb8668fd6a7.png of Lebesgue measure, being unbounded, cannot be integrated with the classical Riemann integral. Therefore spaces of Riemann integrable functions would not be complete in the analysis_c4008d3da4349418a466832c573efdb8668fd6a7.png norm, and the orthogonal decomposition would not apply to them.

Another victory for good ole' Lebesgue!

Orthonormal sets and bases

A set analysis_9fba0f7c9511128704e8cdc3c7b16539475f950b.png in a semi-inner product space analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png is called orthonormal if and only if

analysis_9680c2e123e4916708def89bf813c3c67e766f72.png

For any orthonormal set analysis_37b70b9c3e70e2539ca5741ca72e1311938a0661.png and analysis_43f62a5f9474878c55a1c72b3ae3b6ae15d10906.png,

analysis_5daa2d661b0440ea3fda60a907738747ecf53cb9.png

If analysis_87b5d76d336b9ae7ef7af11997c2dd0a742838b5.png is an orthonormal set in analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png, and analysis_9baff6a5e36526b74c4bf2f9eaa15f44efd8a065.png, where

analysis_ce530d1b05b27c10db3e6d2e65b57997b85f5d35.png

and

analysis_8cd5079d3072b8b4c57598d8763a7573c9831076.png

Then analysis_327f88b6b88b21b430e390264d294fa73d80c279.png and analysis_bab104be5ec1f800be40125311788705725b0668.png for all analysis_28c2ecb2a9689e84c55e0e6e3776f55f2062f9bc.png.

Furthermore,

analysis_316e5551aba32726efddf6f3e9caa4701a478282.png

For any Hilbert space analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png, any orthonormal set analysis_87b5d76d336b9ae7ef7af11997c2dd0a742838b5.png and any analysis_ccb6fc1f06162119ffb348f552d32dbd82303196.png, then

analysis_3fe887cedd81beb4aae505d19db37b6ba7ac9be0.png

analysis_7f43b893b59f7f502d2fd62724b3ee4519074b7d.png: Follows from Bessel's inequality and Parseval-Bessel equality, since we have

analysis_4cd4ebaa9f1296a531f2b67167c58a60c86ae961.png

where the first equality is from Bessel's inequality and the second from Parseval-Bessel equality.

analysis_6081d1ceb52e430c06c79ee9ca315c86854fbd32.png: For each analysis_f293465b753a71ba8064a65b96810f61533d089f.png choose a finite set analysis_e4c75f1975995853316503aebd36a884fd54fda3.png such that analysis_cb359c943cdc0c771118a3195b0f21de8da68e3c.png increases with analysis_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png and

analysis_99683feba26ce1a97c8b2fc6c9fe2fa4b18ff0c5.png

Then

analysis_6dc5c0ec5ca4218ffd92291e945e0993dbea9887.png

is a Cauchy sequence, hence converges to some analysis_43f62a5f9474878c55a1c72b3ae3b6ae15d10906.png since analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png is complete.

Then the net of all partial sums converges to the same limit, concluding our proof.

Let analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png be any inner product space and analysis_23726c31cc6c1501e8a079c2ca7b4e5dfde88913.png be any linearly independent sequenec in analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png.

Then there is an orthonormal sequence analysis_0bc41eced35aff8253161613839e8dc9c9d9b91d.png in analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png s.t. for each analysis_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png, analysis_ee78691e26ba46f5f45ad506c337c6112d648ae8.png and analysis_457aa40efa09d917a1eb46597096d4f548a66a4f.png have the same linear span.

  • Side-note on why Hilbert space > Banach space when talking about bases

    In any vector space analysis_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png, a Hamel basis is a set analysis_37b70b9c3e70e2539ca5741ca72e1311938a0661.png such that every analysis_678b49f7e5ab8f76628d50711bf275b4730960dc.png can be written uniquely as

    analysis_a6977d90cd9cc0f12c10ed7f53b5af7d21467769.png

    with only finitely many analysis_cd31bd9a865d473cab32a9c70a73246ac02ac264.png.

    So, Hamel basis is an algebraic notion, which does not relate to any topology on analysis_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png.

    In a Banach space analysis_f0d6163f2cf3dde6a41bd7e151046bfc48e78a6d.png, an unconditional basis is a collection analysis_37b70b9c3e70e2539ca5741ca72e1311938a0661.png such that for every analysis_678b49f7e5ab8f76628d50711bf275b4730960dc.png,

    analysis_86b15dd365db432ef67c09ec4359acb0974daa0a.png

    converging for analysis_91e92998c43770595cfa6e0f4e74def49e43f939.png.

    In a separable Banach space analysis_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png, a Schauder basis is a sequence analysis_fc4a3eeacd00eeec901f3fd45ab5ad9f28f34fd0.png such that for every analysis_678b49f7e5ab8f76628d50711bf275b4730960dc.png,

    analysis_5933f96c2bae55d5350190e0ae1b0418e6be6ac6.png

    It is possible to find a Schauder basis in the "most useful" separable Banach spaces, but Schauder bases may not be conditional bases, and in general it may be very hard to find unconditional bases.

  • Orthonormal basis for Hilbert space

    Coming up with a basis for an infinite-dimensional space comes down to constructing a sequence of orthonormal vectors analysis_2ce73aaf3979a944b1f3cada28e0297e8973cba7.png by taking some vector for which the projection analysis_acac7bbcaedfe9cee3ac40cbfffd79ab44c3e532.png, i.e. all of analysis_988b807e150272a7d88861d62d63c555893db817.png does not lie in analysis_58e5956e6c0c618290ffdf55ee988f7111ffce83.png.

    analysis_e2888e57d8350da1c811406fde49c0ecc66e7f29.png

    Then we prove that this gives us a space which is dense in the "parent" space.

    More concretely, let analysis_92df7f119b79398e866fd23061f30305aa6b3cb8.png and analysis_7b901a5e7320fd35a99a7bb4ea6d86e6e7ed2ba0.png be defined by choosing some analysis_5abac2f7b37600db005eb76d286903d266a98ae5.png such that analysis_833be61613d63b47dd3707b0dbc121aa019cff53.png, i.e. analysis_bd6286bf7988ee5b964db35b10a8750194284424.png, and

    analysis_33540ac2a8741121d0f497c95e2500b67b5bf840.png

    Every Hilbert space has an orthonormal basis.

    If a collection of orthonormal sets is linearly ordered by inclusion (a chain), then their union is clearly an orthonormal set.

    Thus by Zorn's lemma, let analysis_37b70b9c3e70e2539ca5741ca72e1311938a0661.png be the maximal orthonormal set.

    Take any analysis_43f62a5f9474878c55a1c72b3ae3b6ae15d10906.png. Let

    analysis_628f89f549568e5c7d6e24a2a7fd2e3758caf738.png

    where the sum converges by Bessel's inequality and Riesz-Ficher theorem.

    If analysis_e06b484ee2b1375e0428320e18703e830425f7db.png, we are done. Otherwise, then analysis_51fe2cfa5d5f0a7fbcded8601fb7c482c0875192.png for all analysis_28c2ecb2a9689e84c55e0e6e3776f55f2062f9bc.png, so we can adjoin a new element

    analysis_255268ebb5dfcac389b1d7ecb02da74582ca60eb.png

    contradicting the maximality of the orthonormal set.

    Every Hilbert space is isometric to a space analysis_e8e25c6bd5998d101664680e366d4fc5f77ea047.png for some set analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

    Let analysis_f77d54526242ba5ae107b626d00499c5f8ff6bbe.png be an orthonormal basis for analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png.

    Then analysis_f274f95a66fb1aef976867b0fb697ff46f30fecc.png takes analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png into analysis_e8e25c6bd5998d101664680e366d4fc5f77ea047.png by Bessel's inequality.

    This function preserves inner products by the Parseval-Bessel equality. It is onto analysis_e8e25c6bd5998d101664680e366d4fc5f77ea047.png by the Riesz-Fischer theorem, concluding our proof.

    For any inner product space analysis_f854480bc0d9a8f205ce374efeb6d1488b2dc02e.png, an orthonormal set analysis_37b70b9c3e70e2539ca5741ca72e1311938a0661.png is an orthonormal basis of analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png if and only if its linear span analysis_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png is dense in analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png.

    Let analysis_6c7ea6bdfe94a717547f8acd2e497da1b95fac2d.png be a orthonormal sequence in a Hilbert space analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png.

    The following are equivalent:

    1. If analysis_43f62a5f9474878c55a1c72b3ae3b6ae15d10906.png s.t. analysis_3914afaf7d1f99b2033cd83017bca29135ef13bb.png for all analysis_d8797d487554fc78e460f5843167fcb01b53e76b.png, then

      analysis_595767d48c9abaf056a1554ebc8619768861b5e3.png

      In other words, the sequence analysis_3bb5771b7581008a7f9ab1c68b3499620cc911f8.png is a maximal orthonormal family of vectors.

    2. Span of analysis_ccf3080525fee024a099717cd8c6184d03125a60.png is dense in analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png

      analysis_9ca0f1b527097ad7df6941d730279e6cd85f9c69.png

    3. Unique convergence

      analysis_457bcdd1b1b637c434e6850f00e02b1c4127333a.png

      then

      analysis_a611cab55284e397bebce650d543cc19a0d3ac25.png

    4. Inner product on basis

      analysis_76588e5f7d141e4a17d26168a8ccd4158fd323ff.png

    5. The norm

      analysis_7ea26e9e7a559088f00e0543efc797e3149049e6.png

    If one of these statements hold (and thus all of them hold), we say analysis_6c7ea6bdfe94a717547f8acd2e497da1b95fac2d.png is an orthonormal basis of analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png.

    analysis_72fa90339315b232b5a08b0f69925ad3aaac3aa2.png: Idea is that analysis_ab912040348c31eb2b9433f6e0a1272b45b36160.png implies analysis_8bf6069caa7dc496cc51c5916118e5a30bc71c62.png as a subspace is simply analysis_f6f37fb0f97e3b5e6ba52d4a1fdf665bb21212d2.png, hence the closure is dense in analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png.

    Suppose analysis_0ab66a72ad73bbd89f85dafd2f3acfd5c6d1a686.png.

    Then since analysis_c0efb2a4d967dc62a157d25acec7c8562ba73fa3.png, we have analysis_933d40156ee9762babb45258d260cbe61f14de9a.png and so analysis_bb937ba19f73201c2fc7d037a5a182d5ee3ac7d5.png s.t. analysis_0f776aa4c9d43722116e1a79c50e68bea9ae3e9b.png and analysis_3914afaf7d1f99b2033cd83017bca29135ef13bb.png for all analysis_d8797d487554fc78e460f5843167fcb01b53e76b.png. But this is a contradiction wrt. analysis_ab912040348c31eb2b9433f6e0a1272b45b36160.png, hence we have our proof.

    analysis_f028aa1f6ba267ed573c12372a7b61c9c1bd0cca.png: Let analysis_43f62a5f9474878c55a1c72b3ae3b6ae15d10906.png and since analysis_4a2536e169652232cf7b990ae10fcf81a6ffb843.png.

    analysis_6f9d15ec5fff17957883fce7225333349445df89.png

    Set

    analysis_ab73664d98ec706ad8c66e96bde9145d2d27ee1b.png

    We make the following observations:

    1. analysis_9dbd309e5fdd91882a9a7efd9af594d963a809c0.png and analysis_ac191994db8c7f892047c6be21bf47519a1c4afc.png.
    2. analysis_212af7bd60acfeea70e96639cd7482e269e359d1.png as analysis_d42970d8ef75b6eb77b66c571a7ec127029eaef6.png which tells us that a particular subsequence converges.
    3. analysis_6e834df6782749961236974bd043d35da006a899.png we have analysis_0f5a44193c7e760fdb13be79db422d188f2aabac.png,

      analysis_05cbe120c786e0d2042a6a4e71be7dd386ac3f83.png

      as analysis_d42970d8ef75b6eb77b66c571a7ec127029eaef6.png, since analysis_dd50e29dea383be86928b9950f0ff820539ac2c5.png. Which tells us that all possible subsequences converge.

    Therefore analysis_c8ca1f7dd9d94cfed1b12d3f084c08bea21ffae3.png.

    analysis_e860701c3c562fc123e34806872765c06cb0a463.png:

    analysis_06c26e3b6bdd57e49c8179a8f4be50bfd19c2e4c.png

    were we have used the interchanging of limits on multiple occasions, and in the final equality used the orthogonality of the analysis_656144f995a30e6266bb64d08bec7a71fe900908.png.

    analysis_d1b8b23af26fdfc7937c8783f254d110b3eecf3e.png: Apply 4 with analysis_e06b484ee2b1375e0428320e18703e830425f7db.png.

    analysis_2ed3e7c7ebd34eb2ca8d62984078bf14c62e1fa9.png: Otherwise we would have extra terms for the norm, rather than just the "Fourier" coefficients → contradiction.

Bounded Linear Operators

Notation

Stuff

Let analysis_b8f3bcc04a0c7cdefbcdd96b4640ad78f4fb15fe.png be linear operator and analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and analysis_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png are normed linear spaces.

Then the following are equivalent

  1. analysis_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png is continuous on all of analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png
  2. analysis_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png is continuous at analysis_5c302d31faee1dc741905749ed4f6a8e7454408b.png
  3. analysis_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png is bounded

analysis_f6c0ff3aae15b9374ec53077b4c5325e028d7046.png is seen by observing that for a sequence analysis_97621c3cd88296433cd36d03df345c6772bc32c9.png we have

analysis_18de5b3afd9b4d5ece43563aa5c8aa572e40788f.png

and so analysis_ab912040348c31eb2b9433f6e0a1272b45b36160.png and analysis_39740af1263c8e9d75965873d956f0c0c9bbcab5.png are equivalent.

analysis_9bd17cd585432dfb88213d0d6cb7861206d03a12.png: Suppose analysis_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png is continuous at analysis_5c302d31faee1dc741905749ed4f6a8e7454408b.png. Thus analysis_d76400b96cefe9bbdc4141db3825780299e7d812.png

analysis_8b818be648d4d0c62fe7c23994dbb2031d7ea88e.png

Let analysis_51d8f75f46260190da0b6d2a889f2788010b25bc.png, and let analysis_8b20eeb9aa4cf882f683f4bd3d9e9e5dc14f6886.png and

analysis_7a8939e45efed4bc3f3cf70870e01e0e0337f66b.png

So

analysis_c9eb65ad1aa88cac480033b5024302e0e2f8e042.png

which gives us

analysis_09e80d425d82eb28b296c0630461d267f2caee54.png

Let analysis_5808334780adb536bdcb980855f5fa7afbe51de5.png be vector space of bounded linear operators from analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png to analysis_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png.

Then

analysis_933a7ae28673734119ca2d74555d75481b946848.png

defines a norm on analysis_615f82fd708ef369f723c21580b83aa1cce24dc3.png.

If analysis_9e2f0b75198f1e75ddf417cd3a737985466d6bcf.png, then all linear maps analysis_b8f3bcc04a0c7cdefbcdd96b4640ad78f4fb15fe.png are continuous.

Hence analysis_e869682c89447bdecb9909c58c5e299df303c67f.png.

Let analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png be a Hilbert space, then analysis_ab335d399e25e7eb4a5b8585d0a2ebe3beabcf9e.png.

There exists a conjugate isometric isomorphism

analysis_8d5e39c366fc427629c50a7e1ef259167aa0fa78.png

where

analysis_0296c01852d955b3c52bfa2be479a996d7d580bb.png

T is onto: let

analysis_c82652c718cf3eb77bd76281a5d5cce52054254f.png

Then analysis_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png is a closed subspace of analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png.

Suppose analysis_97ae57f46033d21a45e9c4663bd5fa9e4e043fb3.png, then we can decompose analysis_c0efb2a4d967dc62a157d25acec7c8562ba73fa3.png, then this implies there exists some nonzero analysis_778cbc87cd71a9a27d4e61cd902a37fd4d756e1b.png.

Let analysis_43f62a5f9474878c55a1c72b3ae3b6ae15d10906.png and consider

analysis_c9c94b4b8cef4c3d35a3ee7972e464b4912d6006.png

Then observe that

analysis_e39ce5cd01ea39217b0bd9b5bb9473f851e3a88b.png

which implies that

analysis_3c5f3f171bebb40f27d1415396ea464f5c760974.png

which implies

analysis_5dfd63d3dd341f38496578425ac78461dc6e68b4.png

So for

analysis_3f5e8a45968aa59348da16380dc91cf12ff43426.png

Therefore,

analysis_36ef97f12a77632e76b2bf3423373db0417396d7.png

or equiv,

analysis_b1db089d0174f583132975b8a7139a827e9cdde7.png

analysis_b965e0865a48d1d5cece6150491c1e1209ac5501.png and analysis_b9220649a53eb15759832fe2fc5e3918236ca21f.png

Let

analysis_aab5af7e7491e41f70e858e190e770b6f192326b.png

is the subspace of finite rank operators.

Then for analysis_2b257a023e8483e72202e5958a111e7e41e08089.png and analysis_7c4df801022f6b6722890f0b6bd7ab8b9a34090f.png, there exists analysis_4d3a9583c4550a76655d948b24157f9ade93b9a2.png, i.e. constants which depend on analysis_7a84c9a383f9772338016d101ccc096be06af784.png (functions yo!) s.t.

analysis_1b60a75846059217b9cad9266c7488c731a136ee.png

Then:

  • analysis_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png is linear analysis_bd3cc82fdfe9254196ef619a200cd447f48fd25d.png analysis_f0fffa67cfc7590dfe25ef13f72d7632c9485b08.png are linear for all analysis_d8797d487554fc78e460f5843167fcb01b53e76b.png
  • analysis_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png bounded / cont. analysis_bd3cc82fdfe9254196ef619a200cd447f48fd25d.png analysis_f0fffa67cfc7590dfe25ef13f72d7632c9485b08.png are bounded / cont. for all analysis_d8797d487554fc78e460f5843167fcb01b53e76b.png

If analysis_8c002c9fd1d2f478903963978690e92697c70ea2.png then analysis_60a3d51f7443eae75ebd54d90c1223878a1831de.png, i.e. the dual space of analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png!

If analysis_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png is a Banach space, then analysis_5808334780adb536bdcb980855f5fa7afbe51de5.png is a Banach space.

Examples with analysis_72eebf38c2f8c629bdfaf7b11d217de90fedfb22.png, i.e. analysis_8c002c9fd1d2f478903963978690e92697c70ea2.png

analysis_22b1b1a9586892400fab3e4fb4f38aaa91c76fc4.png for analysis_624d7552486d863b0843d6a8f827307be3acb4cb.png

analysis_cfbab87550ea82f977b9a0f83532addc98d09f67.png

Fix analysis_9817812c3b34bce1a528728381917cef7e7f44e7.png and define

analysis_59ab8d82a4f42ec2bbf4bd2b4764c43b6f282719.png

Then

analysis_2e12a08310e9c4d8d7ea8ea0fd4d6b0e151dfc84.png

by Hölder's inequality. So analysis_69db7206780dd4e28ff6015c866f1ec99b0671d0.png is bounded,

analysis_f1cf489227b6f2b3e34c1022792c787b85e968ba.png

(remember we fixed analysis_eb83d466c7d035356e9f39998f357cee73da1e26.png).

Hence analysis_d064dc2f7b72b5e8a57081cdc43bb46945878648.png and analysis_a97e23af1cb329fbd795fe47c98227d169a23c15.png.

Letting

analysis_65df1863dac97b25b03247ccbdcc4b979c19a7d3.png

Then we attain the UB, and so we have equality.

We can then isometrically embed analysis_21ece0ba649d2ee5b8b44b6d0e72f541a8460ae7.png into analysis_cf55aedc74497aa060cdd3b9a8bbcd5357cf83c4.png. That is, we can show that for any analysis_9997fb2e667c36fb8eb1829061e1462cfcb609ff.png, there exists analysis_b46eba1fa2f222b67993a5fd77661a03df351cc6.png s.t.

analysis_e9d4a1dfc49a3f20f6aa45c1e66c8485ddf106a6.png

Hilbert-Schmidt operators

Let analysis_ee03b7921565c5ef5ef72b9b5b9a15e2a8578155.png where analysis_4984ee810c6bd0ed5859513762ca22885e3f41bf.png are separable Hilbert spaces.

We say that analysis_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png is a Hilbert-Schmidt operator if

analysis_d10e0c70ae868eb45d27f40ba7541caddc7cd335.png

The space of all such operators are denoted analysis_69f23103c9109783a327a79a5c91f1a40d90000d.png with the norm defined

analysis_e0f239e488d1fff8ef49bb717c011db942e5da21.png

analysis_69f23103c9109783a327a79a5c91f1a40d90000d.png is a vector space and

analysis_f68b54b2f28567522e6e2d11698ab2a4e0f24ab9.png

but analysis_69f23103c9109783a327a79a5c91f1a40d90000d.png is not a closed subspace of analysis_221cf9026bd8f78c4a9ef3d126969ecec1248c93.png since

analysis_e523aee0a9ba381be32c2527fbb4af03787a0b6a.png

  • Suppose analysis_617dd66fb0d7bb59bf162234f07d32d3bd8c2ecb.png be a ONB for analysis_1641d18cc980f8db14cdff95d7417a8526eef446.png and analysis_ba25873e5ec27630b4959ccd3c73f8c45b86ff67.png ONB for analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png.
  • Let analysis_9154c3b8ad61f18ed3c856ebcb1be593fac1ff70.png then

    analysis_b8a55de111957a554f4b0c8de6a7ba7fb632ce23.png

    and

    analysis_faf251de60039cb99cf2e528495d6c0ec8c1330f.png

  • Thus,

    analysis_57ea5195d9b0569246459f33259fa3dd27152b1f.png

    (switching sums we can always when all the terms are nonnegative).

  • Hence, the definition of a Hilbert-Schmidt operator (also the norm analysis_e4d94990116a271e6a4a17215a8d1c74aa4f132d.png) is independent of the choice of ONB

Let

analysis_7e94fe0f59a8feabd2358ea32e5480fd86deec55.png

Then analysis_52b30b010b1e06649ef26aa2557bac1f5c8e339b.png is a FR operator.

analysis_e664c26e8896f7b8f1c9bf5eac0abe6d572e7b81.png as analysis_d42970d8ef75b6eb77b66c571a7ec127029eaef6.png then

analysis_8c9428288fca9dee4ed6ab4742851a7189bb0821.png

since then analysis_7a3631d5a97e18c30b81120211fdf3844ab47ab2.png which would imply that banalysis_69f23103c9109783a327a79a5c91f1a40d90000d.png contain the limit points of analysis_0601f0fc57d5ab9277506bf11158235389b065b0.png.

analysis_6478dd33b63a00eed7438a8579f3151400adde57.png

since analysis_9154c3b8ad61f18ed3c856ebcb1be593fac1ff70.png i.e.

analysis_d10e0c70ae868eb45d27f40ba7541caddc7cd335.png

Example: kernel operators on analysis_5616278cb8a851961160c500921111c9f69713b3.png

Let analysis_13045f505e3750d0a43bb088a7b0b4c208901c2c.png be what's called a kernel on analysis_f64c6c1f9f56923793bd78a927b47ea43daf66e5.png and let

analysis_ef9fd1ab0426750896831a6eca4ea3a34926b423.png

called a integral operator of analysis_1641d18cc980f8db14cdff95d7417a8526eef446.png.

Then

analysis_e9ac615b4ca2044a6bcb4c6c822de04d94f3be38.png

which implies that

analysis_bc4c546c6a6bab730ea349e7ed6382291e28a0d3.png

That is,

analysis_da66af439d4f3f53f6e8064d04bc48b62534905c.png

So analysis_693ec4b7194b2efe9256b21a1f8a855c5e198eb2.png is bounded and analysis_ddcbf8a7ac542e4024d83e82370f5312c33f03c4.png.

analysis_b4bea84840159adbc44742e2a9b851fbad45f430.png

Let analysis_cdc43f258de7d2dccadb78deeeaa434434d1acda.png be the completion of analysis_6d0920681f70297711003c92ba6ff47c7cee72a8.png.

Consider the ONB defined by the Fourier coefficients analysis_6e10c13c369149c3306c4561db8f7ae348aa06f1.png.

analysis_6f7f5c66fb8dfc057f5c02a8d9782bc2cf1d5fca.png

where

analysis_4261bece2d63e84a0bb2a9d4f8bb8320ddc90791.png

Summing over analysis_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png

analysis_4c90f5ea1e2154137d55bfaf9cafe8509e1d3bdf.png

Compact operators

Notation
  • analysis_c0d6d9d141779b1f628920a2919ac61ae757b886.png denotes the space of compact operators
  • analysis_e0d94d064d2d143589742b5dd861bee6aac5aef3.png, i.e. denotes the set of all convergent sequences
Stuff

Let analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and analysis_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png be normed spaces and let analysis_b338a64f428bffe6defd377b5f77ef3e65c91d8e.png.

Then analysis_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png is said to be a compact operator if analysis_304da699d5ad02e4e78d0b4d84162ec650b94314.png is a compact subset of analysis_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png.

That is, analysis_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png is compact if the image of the unit closed ball is compact.

Recall that in finite-dimensional space, by Heine Borel, closed and bounded subsets are compact.

Hence finite-rank operators are always compact!

Suppose that analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is a normed space, analysis_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png is a Banach space and that analysis_70dd65adcb90d9c11e1637a507837eebf3fca00d.png is a compact operator.

Suppose there is a analysis_b338a64f428bffe6defd377b5f77ef3e65c91d8e.png s.t. analysis_d24bd81c7f5d3bccc3b1974c1fd3b705f1286d9c.png as analysis_a4c8dc9d99a55084552339a4e055c35b9272ba80.png. Then analysis_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png is compact.

  1. analysis_6b93baa23c0e2520c8e8ce196b0e9c6c8c235e1d.png is a subspace of analysis_5808334780adb536bdcb980855f5fa7afbe51de5.png
  2. If analysis_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png is a Banach space, then analysis_6b93baa23c0e2520c8e8ce196b0e9c6c8c235e1d.png is closed (i.e. Theorem thm:limit-of-compact-operators-is-compact since it contains all its limit points)
  • Supose analysis_4984ee810c6bd0ed5859513762ca22885e3f41bf.png are separable Hilbert spaces (thus there exist bases)
  • We have

    analysis_5de14514a087d280c8d45bc04c700d4677b60a60.png

    • Though we will only prove analysis_48f7fcfdf43129d4f9a4a296a897d2401598e529.png
  • Let analysis_40a7cd169f38abfcc7aadafa52c16958c3a6cb06.png linear and analysis_414153d09a76a972e105f9e8b3262014b0db769b.png be a ONB in analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png, and assume analysis_86878429d0ecb5de6974241ba295543cd2ba7cc1.png since otherwise "containments" would be equality for all in the above.
  • Furthermore, suppose

    analysis_d7ff02db5a906d51452ce8ba56f9b3fc73077fe0.png

  • analysis_49e7e963aad5cc9a3f59f75ed606de36c3098a58.png iff analysis_47904f16961e7b9d0ac1bc84da0cc501bfc0ef15.png and analysis_59fa1c87668bf3c6c8fd3f9091c5f3d6ef74c514.png, since

    analysis_fda6146a154cda977b3ca75f0674ed8886e121e9.png

  • analysis_4d8d2b6b86c29653f54cf24a683f97ea277dba77.png iff analysis_8a49db0c56d3fa23c98639bedea64c4afeb2949d.png and analysis_8ecf795e7a4ac72996d5174601c681ceafe27c1e.png
  • analysis_6ef1d5a74cf7e666175610d41575dd84ba46c096.png iff analysis_dba9e0ea62d0d07bb037c8ab452b21e4169437ea.png
  • analysis_37437f6f387f46b3a622c7643a9ad01465dbbb98.png iff analysis_234a9267d6ef0ed78c65f0923a967bfeca364b37.png
  • analysis_49e7e963aad5cc9a3f59f75ed606de36c3098a58.png iff analysis_8a49db0c56d3fa23c98639bedea64c4afeb2949d.png
  • From this we get the above sequence of containments, since

    analysis_c055d8db8547e12859a556a25e80542f04036323.png

Spectral Theorem for Bounded Self-Adjoint Operators

Notation

  • analysis_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png is the separable complex Hilbert space
  • Operator norm of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png on analysis_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png is

    analysis_d3a24c1fd226998385fec8f51b987756fa0ad313.png

    is finite.

  • Banach space of bounded operators on analysis_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png, wrt. operator norm is denoted analysis_b8df6c238cc24d6136d9771635a522db8da23fdb.png.
  • analysis_bacaf3d072bf37503881b213c59ab985a147c4f7.png denotes the resolvent set of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png
  • analysis_41a71a01903b8fd1d931aa867d5f0c3c4c4a5bf3.png denotes the spectrum of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png
  • analysis_6f0ec29ab78652b52ddfd73d24562ff22c162642.png denotes the projection-valued measure associated with the operator self-adjoint analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png
  • For any projection-vauled measure analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png and analysis_196ddbe8da3548ec409c5b6a8dce31fc9543fba0.png, we have an ordinary (positive) real-valued measure analysis_9f3218af7dda488d97e5a5bad2ce36752723e71f.png given by

    analysis_45f5aeddef60dacc01d05b6c29781cf4dff978d9.png

  • analysis_a7106423e6a132cea5db947e9ecfdd0670890384.png is a map defined by

    analysis_6abe9be4ce10bade2b415398dcf15084214fb0f0.png

  • Spectral subspace for each Borel set analysis_8bb977edb6466538b1ce951d2a306ca45fbdca10.png

    analysis_373d9a36ee8cf6cf00eb42701ffc550b99090aa0.png

    of analysis_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png

  • analysis_09f46c13958381645200ecbcff66fd4e1c27a547.png defines a simultanouesly orthonormal basis for a family analysis_3a0c1b9067643a47e31b9d1663f95fcb4048db79.png of separable Hilbert spaces

Properties of Bounded Operators

  • Linear operator analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png on analysis_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png is said to be bounded if the operator norm of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png

    analysis_d3a24c1fd226998385fec8f51b987756fa0ad313.png

    is finite.

  • Space of bounded operators on analysis_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png forms a Banach space under the operator norm, and we have the inequality

    analysis_561a8ba71c40964399bf3f176c658613db0e73c9.png

    for all bounded operators on analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png and analysis_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png.

For analysis_6f1378740525d29d0b8c054f3b69e304f3802989.png, the resolvent set of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png, denoted analysis_bacaf3d072bf37503881b213c59ab985a147c4f7.png is the set of all analysis_6a8bfebe144986d9a65acc5ce1b692fe4b0ba201.png such that the operator analysis_8aa8508fd75eebefd63ac5fae6ce936ba1a4ba01.png has a bounded inverse.

The spectrum of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png, denoted by analysis_41a71a01903b8fd1d931aa867d5f0c3c4c4a5bf3.png, is the complement in analysis_20a24d1ed449dcbb47090631b2f93ca1189c3c42.png of the resolvent set.

For analysis_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png in the resolvent set of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png, the operator analysis_6a6699eb35f32545e8f4135014afa079b470f30e.png is called the resolvent of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png at analysis_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png.

Alternatively, the resolvent set of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png can be described as the set of analysis_6a8bfebe144986d9a65acc5ce1b692fe4b0ba201.png for which analysis_8aa8508fd75eebefd63ac5fae6ce936ba1a4ba01.png is one-to-one and onto.

For all analysis_6f1378740525d29d0b8c054f3b69e304f3802989.png, the following results hold.

  1. The spectrum analysis_41a71a01903b8fd1d931aa867d5f0c3c4c4a5bf3.png of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is closed, bounded and nonempty subset of analysis_20a24d1ed449dcbb47090631b2f93ca1189c3c42.png.
  2. If analysis_a52b5e19cdde324f29c071d482bdac1303cdfef8.png, then analysis_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png is in the resolvent set of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png

Point 2 in proposition:hall13-quant-7.5 establishes that analysis_41a71a01903b8fd1d931aa867d5f0c3c4c4a5bf3.png is bounded if analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is bounded.

Suppose analysis_6f1378740525d29d0b8c054f3b69e304f3802989.png satisfies analysis_003302cbf4b60db1b6797b412be4667d012e73a6.png.

Then the operator analysis_ceec04ccc2a5efb6b6e2591ac2dd0f80a6533df4.png is invertible, with the inverse given by the following convergent series in analysis_b8df6c238cc24d6136d9771635a522db8da23fdb.png:

analysis_c6ef2912da209c031545825c99f017458d8f7534.png

For all analysis_6f1378740525d29d0b8c054f3b69e304f3802989.png, we have

analysis_6de4c54c2068877096aa4c7abf31d295bdf3ba43.png

Spectral Theorem for Bounded Self-Adjoint Operators

Given a bounded self-adjoint operator analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png, we hope to associate with each Borel set analysis_1be15bdaf9866f25469beab46b89e856207c81a6.png a closed subspace analysis_f2abeffb62ce136d1d611fce9e18ab5401268307.png of analysis_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png, where we think intuitively that analysis_f2abeffb62ce136d1d611fce9e18ab5401268307.png is the closed span of the generalized eigenvectors for analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png with eigenvalues in analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png.

We would expect the following properties of these subspaces:

  1. analysis_0034ea3c2d2e7790dc6fc5d01fa373087764c66b.png and analysis_d8f20735391c69f2f5d3cbef5ab2f9763906e406.png
    • Captures idea that generalized eigenvectors should span analysis_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png
  2. If analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png and analysis_b2db59aeb94cbc1050079892ff07b21b493513b7.png are disjoint, then analysis_4d260d163c1ca5703a26575caa389279f323b315.png
    • Generalized eigenvectors ought to have some sort of orthogonality for distinct eigenvalues (even if not actually in analysis_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png)
  3. For any analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png and analysis_b2db59aeb94cbc1050079892ff07b21b493513b7.png, analysis_174913feba6e8898a472d5701f79a93f0501a81f.png
  4. If analysis_7b717a22c79b7fe462a2f6e59262bcd47c279f96.png are disjoint and analysis_df90acaed9f72be569e8c617ec4f853fdbf46174.png, then

    analysis_ca0ebb49296581806b4125482c87a3ca7557d1a8.png

  5. For any analysis_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png, analysis_f2abeffb62ce136d1d611fce9e18ab5401268307.png is invariant under analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png.
  6. If analysis_5794b4b598546503ac0c75c52678d52924bab06f.png and analysis_8f76cc6c958beef63e111c2d03351b495f1c9be0.png, then

    analysis_8bc214043b20102a6b8f407acd00f146d87c9dfb.png

Projection-Valued measures

For any closed subspace analysis_dae3bc011085c76ed31e2c12477df8c12e2be5da.png, there exists a unique bounded operator analysis_a9090c77ce9916955c745920bf8f134a0932d59d.png such that

analysis_3c5772dc8f9a99be724c5465ef4814314a7d7984.png

where analysis_b3f96c59fbac27960c0cb850c195e991d0d611a4.png is the orthogonal complement.

This operator is called the orthogonal projection onto analysis_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png and it satisfies

analysis_c8d534bfc954eb1cd8309e527e64110d7eae66fa.png

i.e. it's self-adjoint.

One also has the properties

analysis_09e577a3705b6751ab8fc42dc5782c83be94a2ef.png

or equivalently,

analysis_c140603b627f8409f9553db1f1a50a7ba1d7cc8e.png

Conversely, if analysis_a9090c77ce9916955c745920bf8f134a0932d59d.png is any bounded operator on analysis_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png satisfying analysis_79b68e5d6f686efdffcaa316bf75359980f36ae1.png and analysis_108c1dada9d7bfb68c4c0cc80ac783f01f005f6c.png, then analysis_a9090c77ce9916955c745920bf8f134a0932d59d.png is the orthogonal projection onto a closed subspace analysis_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png, where

analysis_43c9697fe40871ccfc7d1a535fdf2e65c40ff473.png

  • Convenient ot describe closed subspaces of analysis_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png in terms of associated orthogonal projection operators
  • Projection operator expresses the first four properties of the spectral subspaces; those properties are similar to those of a measures, so we use the term projection-valued measure

Let analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a set and analysis_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png an analysis_dcb897d0137330758a9675e99ee9ec7f93c4d742.png in analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

A map analysis_a0a5da18bdf68959502bd27fa465cf4fd1638785.png is called a projection-valued measure if the following properties are satisfied:

  1. For each analysis_f933ceed1d0fd23b8d8bcc17b875afcab7246893.png, analysis_aa40c294f2551eb7ac81f321ae78295cc29e6375.png is an orthogonal projection
  2. analysis_a1a5a70f855c26c9d1ddb6800e67fef382077d41.png and analysis_bd2540a215495b47c88ffabdbca250a91b4c7e75.png
  3. If analysis_0ec37729c8e1877c8f8d7b82616cda3c21bdeace.png are disjoint, then for all analysis_8929ebd59d804f0b4681a02e4604c24f91a5b1c5.png, we have

    analysis_8edd46701e07126e9b4b4188706d8a52d70722c0.png

    where the convergence of the sum is in the norm-topology on analysis_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png.

  4. For all analysis_7b5b94b41d5ca2143d494b5c5d4f8dc54de78521.png, we have analysis_d5437014fbddb52800ab3bb2efc605db4be27491.png

Properties 2 and 4 in of a projection-valued measure tells us that if analysis_4070caa01fede33fe80e39178121e1df2177be4e.png and analysis_fa578000a65909574243704c580c4055ed5a76f0.png are disjoint, then

analysis_7a7f4921b3bfe2110f5626699f51bd10eb594746.png

from which it follows that the range of analysis_8245ccebfb693a435aa6f108d9d9dc6070492efe.png and the range of analysis_afeb28fd2597645cb9ae05ae855d3dc7333f47a3.png are perpendicular.

Let analysis_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png be a analysis_dcb897d0137330758a9675e99ee9ec7f93c4d742.png in a set analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and let analysis_741ea1e115f85e1054f975a0d447b2197d65f167.png be a projection-valued measure.

Then there exists a unique linear map, denoted

analysis_ab18cdcf74e5352ed8866baa952744ce7de4a04d.png

from the space of bounded, measurable, complex-valued functions on analysis_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png into analysis_b8df6c238cc24d6136d9771635a522db8da23fdb.png with the property that

analysis_01bd399f2fcbb91a39c3cbf7bd49d3713b014549.png

for all analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png and all analysis_196ddbe8da3548ec409c5b6a8dce31fc9543fba0.png.

This integral has the following properties:

  1. For all analysis_f933ceed1d0fd23b8d8bcc17b875afcab7246893.png, we have

    analysis_af379a420d8c662e6b438ff66e9e578d04f45e71.png

    In particular, the integral of the constant function analysis_4468973182b954eeeb1a22bfe0c5b928511fa9f2.png is analysis_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png.

  2. For all analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png, we have

    analysis_67fe6795a644d0af0a7004acc6f62ce15e32c217.png

  3. Integration is multiplicative: For all analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png and analysis_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png, we have

    analysis_99c4b3a8dadf2ccccc5af1b9014701daf3dc10c6.png

  4. For all analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png, we have

    analysis_92d8d3be3502dbd1359d9552890057b3860ae53a.png

    In particular, if analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is real-valued, then analysis_af6bbcea1e62b5768eda04cc20ea7f2f64d22eb3.png is self-adjoint.

By Property 1 and linearity, integration wrt. analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png has the expected behavior on simple functions. It then follows from Property 2 that the integral of an arbitrary bounded measurable function analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png can be comptued as follows:

  1. Take sequence analysis_dfdd090ed57eac18401044d91801cceb018fdda4.png of simple functions converging uniformly to analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png
  2. The integral of analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is then the limit, in the norm-topology, of the integral of the analysis_dfdd090ed57eac18401044d91801cceb018fdda4.png.

A quadratic form on a Hilbert space analysis_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png is a map analysis_61d679c639259b6b94fc80216239d7f6172c8628.png with the following properties:

  1. analysis_967d8ef613988996b3544790631d6f61800cb25c.png for all analysis_196ddbe8da3548ec409c5b6a8dce31fc9543fba0.png and analysis_6a8bfebe144986d9a65acc5ce1b692fe4b0ba201.png
  2. the map analysis_d5bbcf122da3a01863ca032167777ae5952befb0.png defined by

    analysis_5818fea04112a5c12e71114cf1056826fc355db6.png

is a sesquilinear form.

A quadratic form analysis_9fe46e6ccfd065fb50f2eeb9cbdc65835e9ec321.png is bounded if there eixsts a constant analysis_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png such that

analysis_be605b31d8337da565b44fedc1c3a59d694580bb.png

The smallest such constant analysis_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png is the norm of analysis_9fe46e6ccfd065fb50f2eeb9cbdc65835e9ec321.png.

If analysis_9fe46e6ccfd065fb50f2eeb9cbdc65835e9ec321.png is a bounded quadratic form on analysis_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png, there is a unique analysis_6f1378740525d29d0b8c054f3b69e304f3802989.png such that

analysis_da54f04d317ef544c5a94c1050ebbfc6ff03bbb1.png

If analysis_6065f2b22f6f529ab9631ce6799c1acc8b5f4650.png belongs to analysis_492d525117d0dcc93d066c8759f46b98cf9980ca.png for all analysis_196ddbe8da3548ec409c5b6a8dce31fc9543fba0.png, then the operator analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is self-adjoint.

Spectral Theorem for Bounded Self-Adjoint Operators: direct integral approach

Notation
  • analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png is a analysis_026b58b7d0ddfb732bc0a0630d01a61cc001cc27.png measure on a analysis_dcb897d0137330758a9675e99ee9ec7f93c4d742.png analysis_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png of sets in analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png
  • For each analysis_11d540a6fa79f689fd95ec9cc15f1a16b185f327.png we have a separable Hilbert space analysis_ad92e251486f17ae5d412863e949703d030ab96e.png with inner product analysis_5cf50851931c4ff5b29313fd071a153f4cfd31ba.png
  • Elements of the direct integral are called sections analysis_aa1698cb8ee1665238ec3e91824191643c62ee93.png
Stuff

There are several benefits to this approach compared to the simpler "multiplication operator" approach.

  1. The set analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and the function analysis_5dc59e8e67581e3469d6fd42edc34acb3f86c387.png become canonical:
    • analysis_0754fda19bdabf81a4ee9c8074f7d0e2eb7518e3.png
    • analysis_15ff1fdcb5b3d2a6babc4e830ddddb341c5ad11e.png
  2. The direct integral carries with it a notion of generalized eigenvectors / kets, since the space analysis_ad92e251486f17ae5d412863e949703d030ab96e.png can be thought of as the space of generalized eigenvectors with eigenvalue analysis_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png.
  3. A simple way to classify self-adjoint operators up to unitary equivalence: two self-adjoint operators are unitarily equivalent if and only if their direct integral representations are equivalent in a natural sense.

Elements of the direct integral are called sections analysis_aa1698cb8ee1665238ec3e91824191643c62ee93.png, which are functions on analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png with values in the union of the analysis_ad92e251486f17ae5d412863e949703d030ab96e.png, with property

analysis_3d84f93b5eb82c0dc254de2f53d71bb93ff1413d.png

We define the norm of a section analysis_aa1698cb8ee1665238ec3e91824191643c62ee93.png by the formula

analysis_abb8f82664f4596d837d635bb8ca9b4cf1e3305e.png

provided that the integral on the RHS is finite.

The inner product between two sections analysis_f3527ff67a51b84614a04556c727c038d0a7a4a9.png and analysis_500fcd494df13a53a23eaadcb27199fe812b739b.png (with finite norm) should then be given by the formula

analysis_7cf7a77520496bb7a3141669e6eee41c82383921.png

Seems very much like the differential geometry section we know of.

  • analysis_ad92e251486f17ae5d412863e949703d030ab96e.png is the fibre at each point analysis_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png in the mfd.
  • analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is the mfd.

First we slightly alter the concept of an orthonormal basis. We say a family analysis_3e6ce93c408fe9d4e69bfa680cd40284daf65fe2.png of vectors is an orthonormal basis for a Hilbert space analysis_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png if

analysis_b251db11124517f8526c5eb3710365d2ddcec92a.png

and

analysis_e99a98f43b79cf22e02cde11866e3b572a51d375.png

This just means that we allow some of the vectors in our basis to be zero.

We define a simultanouesly orthonormal basis for a family analysis_3a0c1b9067643a47e31b9d1663f95fcb4048db79.png of separable Hilbert spaces to be a collection analysis_09f46c13958381645200ecbcff66fd4e1c27a547.png of sections with the property that

analysis_3bfc12c9980fd658c433aaffc4e4f7b46fd5e449.png

Provided that the function analysis_645a681c7a3e7c03cdb7683b97799acb64223b3d.png is a measurable function from analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png into analysis_4d2f45153627eceed2f100b44db54377e2c0033b.png, it is possible to choose a simultaneous orthonormal basis analysis_fca298202d9cbc0fff267e972b581c767a8e9729.png such that

analysis_5dcbdf7e585d622ba988829f0662654134875684.png

is measurable for all analysis_d8797d487554fc78e460f5843167fcb01b53e76b.png and analysis_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png.

Choosing a simultaneous orthonormal basis with the property that the function

analysis_ceffe25faec7fd033ecf07006d37be84758ae57e.png

is a measurable function from analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png into analysis_4d2f45153627eceed2f100b44db54377e2c0033b.png, we can define a section to be measurable if the function

analysis_1a9d4158ea58f7f1b68e7a5d3c096bd225348fc4.png

is a measurable complex-valued function for each analysis_d8797d487554fc78e460f5843167fcb01b53e76b.png. This also means that the analysis_53a4bc93f41d7733fb9e4821fdadf52d69030950.png are also measurable sections.

We refer to such a choice of simultaneous orthonormal basis as a measurability structure on the collection analysis_3a0c1b9067643a47e31b9d1663f95fcb4048db79.png.

Given two measurable sections analysis_f3527ff67a51b84614a04556c727c038d0a7a4a9.png and analysis_500fcd494df13a53a23eaadcb27199fe812b739b.png, the function

analysis_137dcf00cafc0c51510ef0d86141544e8bf82a55.png

is also measurable.

Suppose the following structures are given:

  1. a analysis_026b58b7d0ddfb732bc0a0630d01a61cc001cc27.png measure space analysis_35993036aec555a7097e8f33d1a67704e95153bf.png
  2. a collection analysis_dd225d091a1ecdc6b815c3e87ac3def0c038b97c.png of separable Hilbert spaces for which the dimension function is measurable
  3. a measurability structure on analysis_dd225d091a1ecdc6b815c3e87ac3def0c038b97c.png

Then the direct integral of analysis_ad92e251486f17ae5d412863e949703d030ab96e.png wrt. analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png, denoted

analysis_739aeef9a24d3699d121546ebb65ae5cf50381d0.png

is the space of equivalence classes of almost-everywhere-equal measurable sections analysis_aa1698cb8ee1665238ec3e91824191643c62ee93.png for which

analysis_71ff5e135eaa1ea78cb0e8fed6afb680be5d4b1f.png

The inner product analysis_29f3145d892590ef40dd2f24b099e3ebb901a92e.png of two sections analysis_f3527ff67a51b84614a04556c727c038d0a7a4a9.png and analysis_500fcd494df13a53a23eaadcb27199fe812b739b.png is given by the formula

analysis_5268dcad2c0282114fa787261678cfe6dae7bfbf.png

If analysis_6f1378740525d29d0b8c054f3b69e304f3802989.png is self-adjoint, then there exists a σ-finite measure analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png on analysis_41a71a01903b8fd1d931aa867d5f0c3c4c4a5bf3.png, a direct integral

analysis_f491cff699bf9fb193b51135fb7f2eca12d306ab.png

and a unitary map analysis_da9cb51849b13210fa778a80b9907f86fe90c379.png between analysis_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png and the direct integral such that

analysis_a33621a653bbc32f1d088651f15a379343b5f55e.png

for all sections analysis_a66c90b4b6e58e0b6c78ba045513cb27623290ca.png.

Proofs

Notation
  • analysis_6f1378740525d29d0b8c054f3b69e304f3802989.png with spectral radius

    analysis_1ad0084a5a98f689b6ead58822b0ad0a2ed345a5.png

Stage 1: Continuous Functional Calculus
Stage 2: An Operator-Valued Riesz Representation Theorem

Let analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a compact metric space and let analysis_b3ca7489571b2d31006c791334301d8f88436b6f.png denote the space of continuous, real-valued functions on analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

Suppose analysis_ce7fb4eec15bb6f3e1b146dc1f73a301cd6686f5.png is a linear functional with the property that analysis_f89b13726ed6d17890417243b11a3a86724def99.png is non-negative if analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is non-negative.

Then there exists a unique (real-valued, positive) measure analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png on the Borel sigma-algebra in analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png for which

analysis_8ba8b057f37fa0ceaf7b8171442b33ef8cc95980.png

Observe that analysis_acb0106122b7f90d9bd5639367a141a7e53d8327.png is a finite measure, with

analysis_1a8f75cb4f0c986c071c686a803ffb46a7e6a63c.png

where analysis_7d116d2259707306a9272c99ee5b455da5ea6e09.png is the constant function.

Continuous one-parameter groups

analysis_1af46bea1bcf8a48c8d510ae629e07c7474f5efd.png semigroup or strongly continuous one-parameter semigroup

A strongly continuous one-parameter semigroup on a Banach space analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is a map analysis_fd190383688b0a33f2048c6b806a269bf01deac3.png such that

  1. analysis_1a1319bb9cffaac5d8f73ae18c921f9c489c066a.png (i.e. analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png identity operator on analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png
  2. analysis_19b3689d68ca500a6970c5fdc6f0714dc39fa6e7.png we have

    analysis_71204951b3ec2ba2478347e65c5b827de2195fd4.png

  3. analysis_2155d850ce8bed9735540439943acaa2f4772c4d.png we have

    analysis_5f8671d553134cba3e7e212603bb3b401e63f140.png

The first two axioms are algebraic, and state analysis_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png is a representation of a semigroup analysis_b69478ca858b28c16936aea258ed439e91823fae.png, and the last axiom is topological and states that analysis_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png is continuous in the strong operator topology.

Infinitesimal generator

The infinitesimal generator analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png of a strongly continuous semigroup analysis_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png defined by

analysis_f7de0a838383452c8b406bfa7d5bab172944862c.png

whenever the limit exists.

The domain of analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png, denoted analysis_66762cb4a67a65bd850b35f099c07e2ebf326797.png, is the set of analysis_4c91032750887d8dca229b906727af58e9268232.png for which the limit does exist; analysis_66762cb4a67a65bd850b35f099c07e2ebf326797.png is a linera subspace and analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is linear on this domain.

The operator analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is closed, although not necessarily bounded, and the domain analysis_66762cb4a67a65bd850b35f099c07e2ebf326797.png is dense in analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

The storngly continuous semigroup analysis_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png with generator analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is often denoted by the symbol analysis_9a13b692cc009da54631f89513b3afcd6ad18707.png, which is compatible with the notation for matrix exponentials.

Exam prep

May 2017

1

c

If analysis_5371ca5ae9f351819af8950887eedf389fb2b381.png uniformly continuous, and analysis_c9bd8cf9d48add6e1321661b5f5836d31b8a7cb6.png uniformly then analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png is also uniformly continuous.

Given analysis_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png, find analysis_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png such that

analysis_9ea3c905a829c71a758b8fc4037173b7ec58628c.png

In particular analysis_7454993ddc469896e1b60f13d9d82482b70056de.png,

analysis_1259b54e26ea80838702dd982883fdcea26385b9.png

analysis_3496c3af94f5eb7c84f4fd8d8d2ee83929c4f64a.png is uniformly continuous, implies

analysis_9ff8995a9f675aaffd57e23228b4362538e2d174.png

analysis_d2f0f507c03606c477dd9d86dd87ddbdbb499b24.png

Compactness

analysis_a459e955c0c25b1aa5ebf79274441576fb8884a1.png, analysis_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png is closed and analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is compact, then analysis_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png is also compact.

Let analysis_a17bad6f3dda77ea304de37bdba8b0e81241927c.png be any open cover of analysis_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png.

analysis_635120357b8f3fbc3ea237dafb62a3c94edeeaa9.png

analysis_a17bad6f3dda77ea304de37bdba8b0e81241927c.png might not cover whole analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png. analysis_3d2c88fa3c51d62eadf227157ef825dd35c58a48.png is open, therefore

analysis_77964b312bed56b39aa000de706728c845a26e6f.png

is an open cover of analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png. Thus, due to analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png being compact, there exists a finite subcover of the above cover,

analysis_1f94bee69abce4968785658ecbd23050d30c087a.png

covers analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png, then clearly analysis_f97f160d3b5ce81d849364894555c715bd2c1fac.png is a finite cover of analysis_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png, hence analysis_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png is compact, as claimed.

Proving connectedness of a set

  • Best way to prove a set analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is connected is to check if it's path connected
  • Best way to prove a set analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is not connected (disconnected), we use the definition of connectedness:
    • Find 2 open sets analysis_29990d81cfd882592664e66495eef6da4c563fc7.png such that

      analysis_16638077ea64df3e7d9e01b87105752821244ae9.png

      i.e. two sets such that the union covers analysis_e46729bc781c25bbc7120ee2892cc1c0215af7da.png but they share no elements.

Example: analysis_99d745b80865c44015cadc42242c46b161f99c32.png

  • analysis_98f3be46d1362fc482d974f4e3186c5b274ac560.png
  • analysis_3b65fa87e0c9aff598d75c42527344c2017376d0.png
  • analysis_bd5148fb3f1f7619c09a3dd889cfdb56941c8916.png