$H i l b$ 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.

单词	HilbCategoryOfHilbertSpaces
释义	$H i l b$ 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.
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H⁢i⁢l⁢b category of Hilbert spaces

Definition 0.1.

Definition 0.2.

Remark 0.1.

$H i l b$ category of Hilbert spaces