Hilbert Space

Definition

A Hilbert space is a complete Inner Product Space.

Examples

Finite-dimensional spaces

• ${\mathbb{R}}^n$ with $⟨x,y⟩=x\cdot y$
• ${\mathbb{C}}^n$ with $⟨x,y⟩=x\cdot \bar{y}$

Sequence spaces

• l2 Space

Function spaces

• L2 with inner product