PlanetPhysics/Category of Hilbert Spaces

The [[../Cod/|category]] $\mathscr{H}il{b}_f$ of finite-dimensional [[../Hilbert space/|Hilbert spaces]] is defined as the category whose [[../TrivialGroupoid/|objects]] are all finite-dimensional Hilbert spaces ${\mathscr{H}}_f$, and whose [[../TrivialGroupoid/|morphisms]] are linear maps between ${\mathscr{H}}_f$ spaces. The [[../IsomorphicObjectsUnderAnIsomorphism/|isomorphisms]] in $\mathscr{H}il{b}_f$ are all isometric isomorphisms.

Furthermore, one also has the following, [[../PreciseIdea/|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 $\mathscr{H}$, and whose morphisms are linear maps between $\mathscr{H}$ spaces. The isomorphisms in $\mathscr{H}ilb$ are all isometric isomorphisms.

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