📂Hilbert Space

Hilbert Spaces in Functional Analysis

Definition¹

A Hilbert space is a complete inner product space. It is commonly denoted by $H$ and named after Hilbert.

Description

A complete space is a space in which every Cauchy sequence converges. Since Banach spaces are also complete spaces, Hilbert spaces can be described as Banach spaces with an inner product. Examples include:

• Lebesgue spaces $L^{2}$
• $\ell^{2}$ spaces
• Real number space $\mathbb{R}^{n}$
• Complex number space $\mathbb{C}^{n}$

Properties

• Hilbert spaces are uniformly convex
• Shortest vector theorem
• Orthogonal decomposition theorem
• Riesz representation theorem