# Infinite-dimensional vector spaces

## Real functions as infinite-dimensional vectors

### Real functions are vectors

The real function space is a vector space because it is linear in multiplication and addition.

$$g(x)=cf(x)$$

$$h(x)=f(x)+k(x)$$

## Endomorphisms of infinite-dimensional vector spaces

### Endomorphisms on real functions

We start with our vector $$f(x)$$.

$$h(x)=f(x)g(x)$$

The equivalent of the identity matrix is where $$g(x)=1$$.

These are similar to endomorphisms where all off diagonal elements are $$0$$.

#### Differentiation

$$h(x)=\dfrac{\delta }{\delta x}f(x)$$

#### Integration

$$h(x)=\int_{-\infty }^x f(z) dz$$

### Examples of linear operators on real functions

For a function $$v$$ we can define operators $$Ov$$.

Here we consider some examples and their properties.

#### Real multiplication

$$Rv = rf(x)$$

This operator is hermitian. This is equivalent to a finite operator of the form $$rI$$.

#### Multiplication by underlying real number

$$Xv = xf(x)$$

This operator is hermitian. This is equivalent to a finite operator of the form $$M_{ii}=i$$ and $$M_{ij}=0$$.

#### Differentiation

$$Dv = \dfrac{\delta }{\delta x}f(x)$$

While this operator is not hermitian, the following is:

$$-iDv = \dfrac{\delta }{\delta x}[-if(x)]$$

## Forms on infinite-dimensional vector spaces

### Forms on real functions

A form takes two vectors and produces a scalar.

#### Integration as a form

We can use integration to get a bilinear form.

$$\int f(x) g(x) dx$$

If we instead want a sesquilinear form we can instead use:

$$\int \bar {f(x)} g(x) dx$$

### Functionals

Functionals map functions to scalars. They are the $$1$$-forms of infinite-dimensional vector spaces.

If we have a function $$f$$, we can write functional $$J[f]$$.

#### More

We can define neighbourhoods around a function $$f$$. For example, taking $$y$$ to be $$f$$ with infintesimal changes. to each of the values.

The difference between the functional at both points is

$$\delta J=J[y]-J[f]$$

#### Extrema

If

$$\delta J=J[y]-J[f]$$

is the same sign for all y around f, then J has an extremum at f.

### Hilbert space

A complete space with an inner product. That is, a Banach space where the norm is derived from an inner product.

## Calculus of variations

### Functional integration

Integrate over possible functions?

## Sort

### Banach space

A complete normed vector space

### Wave functions

For a vector in hermitian basis, for each eigenvector we have component. wave function is function on ith component.

## Other

### Dirac delta

#### Kronecker delta

The function is: $$\delta_{ij}$$

If $$i=j$$ this is $$1$$. Otherwise it is $$0$$.

We introduced this in linear algebra.

#### Dirac delta

The Dirac delta replaces the Kronecker delta for continuous functions.

That is, we want:

• $$\delta (x\ne 0)=0$$

• $$\delta (0)=+\infty$$

• $$\int_{-\infty }^{\infty}\delta (x)dx=1$$