## You are here

Home$Hilb$ category of Hilbert spaces

## Primary tabs

# $Hilb$ category of Hilbert spaces

###### Definition 0.1.

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.

###### Definition 0.2.

The *category $\mathcal{H}ilb$ 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.

###### Remark 0.1.

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

## Mathematics Subject Classification

46K15*no label found*46C05

*no label found*46C50

*no label found*46C15

*no label found*46E20

*no label found*18-00

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections