User:Anghel/Sandbox

Definition
Let $H$ be a vector space over $\mathbb F \in \set {\R, \C}$. Let $\struct { H, \innerprod \cdot \cdot_H }$ be a inner product space.

Let $d: H \times H \to \R_{\ge 0}$ be the metric induced by the inner product norm $\norm {\,\cdot\,}_H$.

Let $\struct {H, d}$ be a complete metric space.

Then $H$ is called a Hilbert space.

Definition
Let $H$ be a vector space over $\mathbb F \in \set {\R, \C}$. Let $\struct {H, \norm {\,\cdot\,}_H}$ be a Banach space with norm $\norm {\,\cdot\,}_H : H \to \R_{\ge 0}$.

Let $H$ have an inner product $\innerprod \cdot \cdot_H : H \times H \to \C$ such that $\norm {\,\cdot\,}_H$ is the inner product norm.

Then $H$ is called a Hilbert space.