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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