Definition:Isomorphism (Hilbert Spaces)

Definition
Let $H, K$ be Hilbert spaces.

Denote by $\left\langle{\cdot, \cdot}\right\rangle_H$ and $\left\langle{\cdot, \cdot}\right\rangle_K$ their respective inner products.

An isomorphism between $H$ and $K$ is a map $U: H \to K$, such that:


 * $(1): \qquad U$ is a linear map
 * $(2): \qquad U$ is surjective
 * $(3): \qquad \forall g,h \in H: \left\langle{g, h}\right\rangle_H = \left\langle{Ug, Uh}\right\rangle_K$

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.