\begin{definition}[Definition:Hilbert Space]
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'''.
{{refactor|The following need to be established by the technique of equivalent definitions with an equivalence proof.|level = advanced}}
The '''Hilbert space''' $V$ may be considered as one of the following:
:The complete inner product space $\struct {V, \innerprod \cdot \cdot_V}$
:The Banach space $\struct {V, \norm {\,\cdot\,}_V}$
{{refactor|These can be established by the technique of iff proofs, as AFAIK you would not use them as definitions of $\HH$. If I'm wrong, then make another definition with equivalence proof|level = advanced}}
:The topological space $\struct {V, \tau_d}$ where $\tau_d$ is the topology induced by $d$
:The vector space $\struct {V, +, \circ}_{\Bbb F}$
That is to say, all theorems and definitions for these types of spaces directly carry over to all Hilbert spaces.
\end{definition}