# PlanetPhysics/Category of Hilbert Spaces

The *category* of finite-dimensional Hilbert spaces is defined as the category whose objects are all finite-dimensional Hilbert spaces , and whose morphisms are linear maps between spaces.
The *isomorphisms* in 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 , and whose morphisms are linear maps between spaces.
The *isomorphisms* in are all isometric isomorphisms.

The category of Hilbert spaces has direct sums and is a Cartesian category.