Definition:Isomorphism (Hilbert Spaces)

This page is about Isomorphism in the context of Hilbert Space. For other uses, see Isomorphism.

Definition

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.

The word isomorphism derives from the Greek morphe (μορφή) meaning form or structure, with the prefix iso- meaning equal.

Thus isomorphism means equal structure.