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 $$\mathcal{Hilb$$ 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.