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)
i.e. Banach space is a complete normed vector space.
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).
A Hilbert space is a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product.
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
1.1. Parallelogram Law in Hilbert Space
In Hilbert space following identity holds:
\begin{equation*} || u + v ||^2 + || u - v ||^2 = 2 ( ||u||^2 + ||v||^2) \end{equation*}And every banach space where this identity holds is a Hilbert Space. The inner product can be defined as (known as Polarization Identity):
\begin{equation*} \langle u, v \rangle = \frac 1 4 ( ||u + v||^2 - ||u - v||^2) \end{equation*}