Category:Definitions/Examples of Hilbert Spaces

This category contains definitions of examples of Hilbert Space.

Let $H$ be a vector space over $\mathbb F \in \set {\R, \C}$.

Definition 1

Let $\struct {H, \innerprod \cdot \cdot_H}$ be an 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 a Hilbert space over $\mathbb F$.

Definition 2

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 the inner product norm is equivalent to the norm $\norm {\,\cdot\,}_H$.

Then $H$ is a Hilbert space over $\mathbb F$.