# $Hilb$ 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.

Title $Hilb$ category of Hilbert spaces HilbCategoryOfHilbertSpaces 2013-03-22 18:25:10 2013-03-22 18:25:10 bci1 (20947) bci1 (20947) 10 bci1 (20947) Definition msc 46K15 msc 46C05 msc 46C50 msc 46C15 msc 46E20 msc 18-00 $Hilb$ DirectSumOfHilbertSpaces ClassificationOfHilbertSpaces IndexOfCategories isomorphisms in $Hilb$ Hilbert space morphisms