# Functional Analysis

## 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:
• 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

• be a vector space over and be its dual space
• be the vector space of functions 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 :

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 !