Definition 0.1. The category ℋilbf of finite-dimensional
Hilbert spaceshttps://physicslibrary.org/encyclopedia/NormInducedByInnerProduct.html is
defined as the category whose objects are all finite-dimensional Hilbert spaces ℋf, and whose
morphisms are linear maps between ℋf spaces. The isomorphisms in ℋilbf are all isometric
isomorphisms.
Furthermore, one also has the following, general definition for any Hilbert space.
Definition 0.2. The category ℋilb of Hilbert spaces is defined as the category whose objects
are all Hilbert spaces ℋ, and whose morphisms are linear maps between ℋ spaces. The
isomorphisms in ℋilb are all isometric isomorphisms.
Remark 0.1.
The category of ℋilb Hilbert spaces has direct sums and is a Cartesian category.
