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)