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:

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

Eigenvalues and eigenvectors of infinite-dimensional vectors

Spectral theorem for infinite-dimensional vector spaces

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.

Functional derivatives

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

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:

  • \(\delta (x\ne 0)=0\)

  • \(\delta (0)=+\infty\)

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