\begin{definition}[Definition:Isometry (Hilbert Spaces)]
Let $H, K$ be Hilbert spaces, and denote by $\innerprod \cdot \cdot_H$ and $\innerprod \cdot \cdot_K$ their respective inner products.
A linear map $U: H \to K$ is called an '''isometry''' {{iff}}:
:$\forall g,h \in H: \innerprod g h_H = \innerprod {U g} {U h}_K$
{{explain|Notation $U g, U h$ may need to be amended so as to make it clear that it means the application of the mapping $U$ on both of $g$ and $h$. If this is indeed what it means (as is probable) then suggest $\map U g$ and $\map U h$.}}
\end{definition}