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 SpaceEdit

An inner product space or IPS is a vector space V over a field F with a function   called an inner product that adheres to three axioms.

1. Conjugate symmetry:   for all  . Note that if the field   is   then we just have symmetry.

2. Linearity of the first entry:   and   for all   and  .

3. Positive definateness:   for all   and   iff  .

Definition of a Hilbert SpaceEdit

A Hilbert Space is an inner product space that is complete with respect to its inferred metric.

ExerciseEdit

Prove that an inner product has a naturally associated metric and so all IPSs are metric spaces.

ExampleEdit

  is a Hilbert space where its points are infinite sequences   on I, the unit interval such that

 

converges and is a Hilbert space with the inner product  .

Characterisation TheoremEdit

There is one separable Hilbert space up to homeomorphism and it is  .

ExercisesEdit

(under construction)

Topology
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