# Topology/Hilbert Spaces

A *Hilbert space* is a type of vector space that is complete and is of key use in functional analysis. It is a more specific kind of Banach space.

## Definition of Inner Product Space[edit]

An *inner product space* or IPS is a vector space V over a field F with a function called an inner product that adheres to three axioms.

1. Conjugate symmetry: for all . Note that if the field is then we just have symmetry.

2. Linearity of the first entry: and for all and .

3. Positive definateness: for all and iff .

## Definition of a Hilbert Space[edit]

A Hilbert Space is an inner product space that is complete with respect to its inferred metric.

## Exercise[edit]

Prove that an inner product has a naturally associated metric and so all IPSs are metric spaces.

## Example[edit]

is a Hilbert space where its points are infinite sequences on I, the unit interval such that

converges and is a Hilbert space with the inner product .

## Characterisation Theorem[edit]

There is one separable Hilbert space up to homeomorphism and it is .

## Exercises[edit]

(under construction)