📂Hilbert Space

Hilbert Space in Functional Analysis

Definition¹

A complete inner product space is called a Hilbert space. It is usually denoted by $H$, after Hilbert’s name.

Explanation

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

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

Properties

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