# 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 $\langle \cdot ,\cdot \rangle :V\times V\to F$  called an inner product that adheres to three axioms.

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

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

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

## 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

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

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

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

## Characterisation Theorem

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

## Exercises

(under construction)

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