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 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 Space
A Hilbert Space is an inner product space that is complete with respect to its inferred metric.
Prove that an inner product has a naturally associated metric and so all IPSs are metric spaces.
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 .
There is one separable Hilbert space up to homeomorphism and it is .