category of Hilbert spaces | Physics Library

Definition 0.1 The category $\mathcal{H}ilb_f$ of finite-dimensional Hilbert spaceshttp://physicslibrary.org/encyclopedia/NormInducedByInnerProduct.html 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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