Definition:Hilbert Space/Definition 1

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