Topology/Hilbert Spaces

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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 $⟨\cdot ,\cdot ⟩:V\times V\to F$ called an inner product that adheres to three axioms.

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

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

3. Positive definateness: $⟨x,y⟩\ge 0$ for all $x,y\in V$ and $⟨x,y⟩=0$ 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

$\sum _{i=1}^{\mathrm{\infty }}{a}_i^2$

converges and is a Hilbert space with the inner product $⟨(x_n),(y_n)⟩=\sum _{i=1}^{\mathrm{\infty }}{x}_i\overline{y_i}$.

Characterisation Theorem

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

Exercises

(under construction)

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