ProofWiki:Sandbox/Definition:Hilbert Space

Definition
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 inner product metric.

If $\struct {V, d}$ is a complete metric space, $V$ is said to be a Hilbert space.

Standard Notation
In most of the literature, when studying a Hilbert space $\HH$, unless specified otherwise, it is understood that:


 * $\innerprod \cdot \cdot$ or $\innerprod \cdot \cdot_\HH$ denotes the inner product on $\HH$
 * $\norm {\,\cdot\,}$ or $\norm {\,\cdot\,}_\HH$ denotes the inner product norm on $\HH$

where the subscripts serve to emphasize the space $\HH$ when considering multiple Hilbert spaces.

Also see

 * Definition:Banach Space
 * Hilbert Space with Inner Product Norm is Banach Space