Functional Analysis

Table of Contents

Notation

Definitions

Bounded operator

We say a linear operator functional_analysis_2e030f1ad8ed697cdd8a012e20c9ef316bdaef36.png is bounded if

functional_analysis_61b506c8a6d18cf28420c8b4a63e73adbf0c1461.png

If functional_analysis_a20bca6b2b1dca2278ae5b7fbdc04932be035b8f.png and functional_analysis_8f593a0a71693446eccc9f47248035f8cecbe561.png are normed spaces, a linear map functional_analysis_e0b095c718fe8edcfa2d85aed6555cf654f9dab1.png is bounded if

functional_analysis_6abb95db6c56471a3a6f39a38be6a984a07c7eb3.png

If functional_analysis_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png is bounded, then the supremum above is called the operator norm of functional_analysis_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png, denoted functional_analysis_2e29b4c13a282329b04c399ed9925c03fd5f5d7d.png.

Let functional_analysis_a20bca6b2b1dca2278ae5b7fbdc04932be035b8f.png be a normed space and functional_analysis_8f593a0a71693446eccc9f47248035f8cecbe561.png be a Banach space.

Suppose functional_analysis_f53cf2c3985c80987bd9707fd5a0b28594db886f.png is a dense subspace of functional_analysis_a20bca6b2b1dca2278ae5b7fbdc04932be035b8f.png and functional_analysis_c34c87c0298deef8f43273ceb2336a6e0dc2c0cb.png is a bounded linear operator.

Then there exists a unique bounded linear map functional_analysis_064cfaebbbfd0d1cb8684d135047d30e79bfe4cd.png such that

functional_analysis_d288ae4de69e0a6e0bc4ae9fc6b37cb1c630a17c.png

Furthermore,

functional_analysis_7970da2692251fe180a71c4d2a6fc72546a1341f.png

Theorems

Riesz representation theorem

Notation

  • functional_analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png denotes a Hilbert space

Theorem

This theorem establishes an important connection between a Hilbert space and its continuous dual space.

If the underlying field is functional_analysis_492d525117d0dcc93d066c8759f46b98cf9980ca.png, then the Hilbert space is isometrically isomorphic to its dual space; if it's functional_analysis_20a24d1ed449dcbb47090631b2f93ca1189c3c42.png, then the Hilbert space is isometrically anti-isomorphic to the dual space.

Let functional_analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png be the Hilbert space, and functional_analysis_a3fb56d2270ef93a77460acfeae7f2a78b7ffced.png denote its dual space, consisting of all continous linear functionals from functional_analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png into the field functional_analysis_492d525117d0dcc93d066c8759f46b98cf9980ca.png or functional_analysis_20a24d1ed449dcbb47090631b2f93ca1189c3c42.png.

If functional_analysis_7494375cd141e688509a55711b7f8388c5052bea.png, then the functional functional_analysis_a5eb01df0fc07cd3c0da2b723bc06895b56068e8.png is defined by:

functional_analysis_e295c5fa34fd17827cc2374c28b1eee73365a85d.png

then functional_analysis_6cf3ba438c134327acd3e12c3187add61e7557d2.png.

The Riesz representation theorem states that every element of functional_analysis_a3fb56d2270ef93a77460acfeae7f2a78b7ffced.png can be written uniquely in this form.

Given any continuous functional functional_analysis_a88c4868fbd16d8a34e1a784e268a56bea7b75d8.png, the corresponding element functional_analysis_656379462a0acd1da6acb6b8e46afd7b70d41c84.png can be constructed uniquely by

functional_analysis_a2646748bc94f388dbbd0052199d37a7dd2ed8e3.png

where functional_analysis_127cb270e3d9d0f41354146d98652b6ee5ee8b95.png is an orthonormal basis of functional_analysis_14a40f189f2ce698341c03cd5c099a337431c49f.png, and the value functional_analysis_5dc59e8e67581e3469d6fd42edc34acb3f86c387.png does not vary by choice of basis.

Theorem: The mapping functional_analysis_3599b07bb72f7c38dc352c2af47e3cbffa6af403.png defined by functional_analysis_ba5b3b505cba290be916e5f3bd26314bf5867fe6.png is an isometric (anti-) isomorphism, meaning that:

  • functional_analysis_05594e8ce8f52728892c4c673d8bc28149b4b24d.png is bijective
  • The norms of functional_analysis_93369077affb352dbdb97e8b3182fd50784f2b14.png and functional_analysis_a5eb01df0fc07cd3c0da2b723bc06895b56068e8.png agree: functional_analysis_e3f803ac922dd026ba873d9f7a3ce67b079f30e5.png
  • functional_analysis_05594e8ce8f52728892c4c673d8bc28149b4b24d.png is additive: functional_analysis_243610f1be7268f465bdf98086fcdddbe6daafc0.png
  • If the base field is functional_analysis_492d525117d0dcc93d066c8759f46b98cf9980ca.png, then functional_analysis_d3d5a9d660df3998961e31ab3f79371c6df15f8b.png for functional_analysis_b8e21f665a3fa77a15d59f0bdd468edb1b09dc47.png
  • If the base field is functional_analysis_20a24d1ed449dcbb47090631b2f93ca1189c3c42.png, then functional_analysis_4b8316b55bfde8a930592af394933c16e43a6aa1.png for functional_analysis_66502043999ff4da05f829ddb424bf53e818cf77.png

In the mathematical treatment of quantum mechanics, the theorem can be seen as a justification for the popular bra-ket notation. The theorem says that, every bra functional_analysis_9ca70e3a799d3a6dc5df67f3fae138ac96f0d53e.png has a corresponding ket functional_analysis_fed715cf5dfda1186c93ca7bf08e2a8c4be01a55.png, and the latter is unique.

When we say the dual space is continuous we mean that the linear operator acting on the functions (elements) in the Hilbert space

If functional_analysis_f3771cafedae120e5569c2845958c3e8409a9b92.png is a bounded linear functional, then there exists a unique functional_analysis_13dc4cfcd652fc796f4174036b7702c21e046a88.png such that

functional_analysis_fd7cfe272f615376806622ba2dc3f2b37aa694bc.png

Furthermore, the operator norm of functional_analysis_42b688f7ebfaa0e3791010e7725767cadf2ad420.png as a linear functional is equal to the norm of functional_analysis_4ef26ff1bd3f789ae16862cbfc47f13a48b0f7eb.png as an element of functional_analysis_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png.

The operator norm is defined as

functional_analysis_03c6ca7cc7cdcc3285819f2b5a2c3460fcafc521.png

Mercer's Theorem

Let functional_analysis_ce8a4ce3819875416474aa2dcf1372ccf5b78fbd.png be a symmetric continuous function, often called a kernel.

functional_analysis_1641d18cc980f8db14cdff95d7417a8526eef446.png is said to be non-negative definite (or positive semi-definite) if and only if

functional_analysis_4ecacfbb519598ddb50c6760b53bdd3d3d208703.png

for all fininte sequences of points functional_analysis_09b35293fe0907cef00c439b793448e7e37b0926.png and all choices of real numbers functional_analysis_35118d99a974a7a36d09cab3dd36c44ba1e593c2.png.

We associate with functional_analysis_1641d18cc980f8db14cdff95d7417a8526eef446.png a linear operator functional_analysis_20b144bf5d2c32b9f4833c62e4541eeefe3362c9.png by

functional_analysis_b3bbaddf17cba9622a10a6293be4a7a06437878c.png

The theorem then states that there is an orthonormal basis functional_analysis_127cb270e3d9d0f41354146d98652b6ee5ee8b95.png of functional_analysis_f0c94b7d2b5f0f0c4476ee6445739f9887e06e35.png consisting for eigenfunctions of functional_analysis_693ec4b7194b2efe9256b21a1f8a855c5e198eb2.png such that the corresponding sequence of eigenvalues functional_analysis_db703d5499816e76f608b8777b0088756ba6da45.png is nonnegative.

The eigenfunctions corresponding to non-zero eigenvalues are continuous on functional_analysis_48286279ea8482e9b623863252fcc0f2c6430f89.png and functional_analysis_1641d18cc980f8db14cdff95d7417a8526eef446.png has the representation

functional_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. functional_analysis_1117bfc670de39be88cd023892ec7fc810e8bfe6.png on any compact Hausdorff space functional_analysis_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

Generalized functions / distributions

Notation

  • functional_analysis_8f0bee27c9815106368cf402ae6001c5a8b01ceb.png be a vector space over functional_analysis_492d525117d0dcc93d066c8759f46b98cf9980ca.png and functional_analysis_4ed0af83f29dd43c3335fdaa3eb1bd82c88943f6.png be its dual space
  • functional_analysis_336cc4816ec0737847752cbcbdf529fc618cae33.png be the vector space functional_analysis_e55df2649bdadb7ca51cdc1aeaf3afc12cfb98f1.png of functions functional_analysis_47a6fa85d448d3f72d86d6f87b8adbdf00a7bd04.png with compact support which are infinitely differentiable

Stuff

  • 'singular functions' occur as a rule only in intermediate stages of solut
  • Given a 'singular function' we know the result of its integration against a "good" function functional_analysis_078b85cd3478400338e3a1ee425c2a468644be7e.png:

    functional_analysis_f4ba083b66046f1b3538a6429e3635d84d9721e0.png

We say the sequence functional_analysis_6f800e8020f8ca02c4e110fc5eb1ac0e8111039c.png with functional_analysis_58727f9563abf5cf094e23e77f3ca543aff6ecf3.png converges to zero if

functional_analysis_001aae524a2e5c17d9ee6ec15db50432a9130456.png

i.e. functional_analysis_6026c03e845e7bf21559f8d9abba4def5de3e62f.png has bounded support, and functional_analysis_6026c03e845e7bf21559f8d9abba4def5de3e62f.png and its derivatives converges uniformly to zero.

We say that the linear function functional_analysis_5e6a9de5b9242d6c511f415767a27ace0d08e16a.png is continuous if

functional_analysis_e377c81c8de59db8d158e832237f0e85d8ee3343.png

The set of continuous linear functionals is denoted functional_analysis_1e7fe6ce4982aedb9b7944f5d240c880a8ef3248.png and its elements are called distributions or generalized functions.

An example is the Dirac delta function functional_analysis_87351be92fff407e73907e8fa6418c9738b34548.png defined by

functional_analysis_bdd9ca470ac293d73d907aa6858a2677b503388c.png

We can then define scalar multiplication over functional_analysis_1e7fe6ce4982aedb9b7944f5d240c880a8ef3248.png by functions functional_analysis_85df0037354bfcb1abf5e47316db97ce02940edb.png, then if functional_analysis_2587eb6c79f769a62a987cb1a486a22747e9e11d.png and functional_analysis_6c3c8161dc83728281dcd04ece5a0f620479d122.png, then

functional_analysis_ddc600f46898cc9b3c1d60c1e4eba499e1b5cab0.png

Since we can add distributions, we can construct linear combinations

functional_analysis_e4d2e2d983e719fcd845c39c13aafa8636589207.png

of distributions with coefficients in the ring of functional_analysis_525713c706a073850c6cb733b94e50fceeee3790.png functions. Hence we have a module over functional_analysis_fc49308a4031b1f0e2149b2fdaeb7b5239908934.png!