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)