# Topology/Hilbert Spaces

 Topology ← Banach Spaces Hilbert Spaces Free group and presentation of a group →

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

An inner product space or IPS is a vector space V over a field F with a function ${\displaystyle \langle \cdot ,\cdot \rangle :V\times V\to F}$  called an inner product that adheres to three axioms.

1. Conjugate symmetry: ${\displaystyle \langle x,y\rangle ={\overline {\langle y,x\rangle }}}$  for all ${\displaystyle x,y\in V}$ . Note that if the field ${\displaystyle F}$  is ${\displaystyle \mathbb {R} }$  then we just have symmetry.

2. Linearity of the first entry: ${\displaystyle \langle ax,y\rangle =a\langle x,y\rangle }$  and ${\displaystyle \langle x+y,z\rangle =\langle x,z\rangle +\langle y,z\rangle }$  for all ${\displaystyle x,y,z\in V}$  and ${\displaystyle a\in F}$ .

3. Positive definateness: ${\displaystyle \langle x,y\rangle \geq 0}$  for all ${\displaystyle x,y\in V}$  and ${\displaystyle \langle x,y\rangle =0}$  iff ${\displaystyle x=y}$ .

## Definition of a Hilbert Space

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

## Exercise

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

## Example

${\displaystyle \ell ^{2}}$  is a Hilbert space where its points are infinite sequences ${\displaystyle (a_{n})}$  on I, the unit interval such that

${\displaystyle \sum _{i=1}^{\infty }a_{i}^{2}}$

converges and is a Hilbert space with the inner product ${\displaystyle \langle (x_{n}),(y_{n})\rangle =\sum _{i=1}^{\infty }x_{i}{\overline {y_{i}}}}$ .

## Characterisation Theorem

There is one separable Hilbert space up to homeomorphism and it is ${\displaystyle \ell ^{2}}$ .

## Exercises

(under construction)

 Topology ← Banach Spaces Hilbert Spaces Free group and presentation of a group →