Category:Hilbert Spaces

Let $V$ be an inner product space over $\Bbb F \in \set {\R, \C}$.
Let $d: V \times V \to \R_{\ge 0}$ be the metric induced by the inner product norm $\norm {\,\cdot\,}_V$.
If $\struct {V, d}$ is a complete metric space, $V$ is said to be a Hilbert space.