Banach Space
Table of Contents
A Banach space is a vector space
- with a metric that can compute vector length, and distance between vectors
- and is complete (i.e. every Cauchy Sequence converges)
1. Hilbert Space
A hilbert space is a banach space where the metric/norm comes from an inner product. (i.e. the metric has to satisfy parallelogram law)
E.g. If supremum (i.e. \(L_\infty\)) is the norm of the space then the space is banach but not Hilbert.
Hilbert Space generalize the familiar notions of \(\mathbb{R}^n\) to infinite dimensions. Because of inner product, Hilbert Space have concept of
- orthnormal basis
- projection
- angle
- eignenvectors and eigenvalues
