\begin{definition}[Definition:Isomorphism (Hilbert Spaces)]
Let $H, K$ be Hilbert spaces.
Denote by $\innerprod \cdot \cdot_H$ and $\innerprod \cdot \cdot_K$ their respective inner products.
An '''isomorphism''' between $H$ and $K$ is a map $U: H \to K$, such that:
:$(1): \quad U$ is a linear map
:$(2): \quad U$ is surjective
:$(3): \quad \forall g, h \in H: \innerprod g h_H = \innerprod {U g} {U h}_K$
These three requirements may be summarized by stating that $U$ be a surjective isometry.
Furthermore, Surjection that Preserves Inner Product is Linear shows that requirement $(1)$ is superfluous.
If such an '''isomorphism''' $U$ exists, $H$ and $K$ are said to be '''isomorphic'''.
As the name '''isomorphism''' suggests, Hilbert Space Isomorphism is Equivalence Relation.
\end{definition}