# Functional Analysis

## Table of Contents

## Notation

- represents the Hilbert space over

## Definitions

### Bounded operator

We say a linear operator is **bounded** if

If and are normed spaces, a *linear map* is **bounded** if

If is **bounded**, then the supremum above is called the **operator norm** of , denoted .

Let be a normed space and be a Banach space.

Suppose is a dense subspace of and is a bounded linear operator.

Then there exists a *unique* bounded linear map such that

Furthermore,

## Theorems

### Riesz representation theorem

#### Notation

- 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 , then the Hilbert space is *isometrically isomorphic* to its dual space; if it's , then the Hilbert space is *isometrically anti-isomorphic* to the dual space.

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

If , then the functional is defined by:

then .

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

Given any continuous functional , the corresponding element can be constructed uniquely by

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

**Theorem:** The mapping defined by is an isometric (anti-) isomorphism, meaning that:

- is
*bijective* - The norms of and agree:
- is additive:
- If the base field is , then for
- If the base field is , then for

*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 has a corresponding ket , 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 is a *bounded linear functional*, then there exists a unique such that

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

The **operator norm** is defined as

### Mercer's Theorem

Let be a *symmetric continuous* function, often called a *kernel*.

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

for all fininte sequences of points and all choices of real numbers .

We associate with a linear operator by

The theorem then states that there is an *orthonormal basis* of consisting for *eigenfunctions* of such that the corresponding sequence of eigenvalues is *nonnegative*.

The eigenfunctions corresponding to non-zero eigenvalues are continuous on and has the representation

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. on any compact Hausdorff space .

## Generalized functions / distributions

### Notation

### 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 :

We say the sequence with **converges to zero** if

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

We say that the linear function is **continuous** if

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

An example is the Dirac delta function defined by

We can then define scalar multiplication over by functions , then if and , then

Since we can add distributions, we can construct linear combinations

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