# PlanetPhysics/Category of Hilbert Spaces

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

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

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

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