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

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)