PlanetPhysics/Category of Hilbert Spaces

The category ${\mathcal {H}}ilb_{f}$ of finite-dimensional Hilbert spaces is defined as the category whose objects are all finite-dimensional Hilbert spaces ${\mathcal {H}}_{f}$, and whose morphisms are linear maps between ${\mathcal {H}}_{f}$ spaces. The isomorphisms in ${\mathcal {H}}ilb_{f}$ are all isometric isomorphisms.

Furthermore, one also has the following, general definition for any Hilbert space.

The category Failed to parse (syntax error): {\displaystyle \mathcal{Hilb} of Hilbert spaces} is defined as the category whose objects are all Hilbert spaces ${\mathcal {H}}$, and whose morphisms are linear maps between ${\mathcal {H}}$ spaces. The isomorphisms in ${\mathcal {H}}ilb$ are all isometric isomorphisms.

The category of ${\mathcal {H}}ilb$ Hilbert spaces has direct sums and is a Cartesian category.