$Hilb$ category of Hilbert spaces

Definition 0.1.

The category $\mathrm{H}\mathit{}i\mathit{}l\mathit{}{b}_f$ of finite-dimensional Hilbert spaces is defined as the category whose objects are all finite-dimensional Hilbert spaces ${\mathscr{H}}_f$, and whose morphisms are linear maps between ${\mathscr{H}}_f$ spaces. The isomorphisms in $\mathscr{H}il{b}_f$ are all isometric isomorphisms.

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

Definition 0.2.

The category $\mathrm{H}\mathit{}i\mathit{}l\mathit{}b$ of Hilbert spaces is defined as the category whose objects are all Hilbert spaces $\mathscr{H}$, and whose morphisms are linear maps between $\mathscr{H}$ spaces. The isomorphisms in $\mathscr{H}ilb$ are all isometric isomorphisms.

Remark 0.1.

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

Title	$Hilb$ category of Hilbert spaces
Canonical name	HilbCategoryOfHilbertSpaces
Date of creation	2013-03-22 18:25:10
Last modified on	2013-03-22 18:25:10
Owner	bci1 (20947)
Last modified by	bci1 (20947)
Numerical id	10
Author	bci1 (20947)
Entry type	Definition
Classification	msc 46K15
Classification	msc 46C05
Classification	msc 46C50
Classification	msc 46C15
Classification	msc 46E20
Classification	msc 18-00
Synonym	$Hilb$
Related topic	DirectSumOfHilbertSpaces
Related topic	ClassificationOfHilbertSpaces
Related topic	IndexOfCategories
Defines	isomorphisms in $Hilb$
Defines	Hilbert space morphisms