category of Hilbert spaces
Definition 0.1
The
category
of finite-dimensional Hilbert spaces is defined as the category whose objects are all finite-dimensional Hilbert spaces

, and whose morphisms are linear maps between

spaces.
The
isomorphisms in

are all isometric isomorphisms.
Furthermore, one also has the following, general definition
for any Hilbert space.
Definition 0.2
The
category
of Hilbert spaces is defined as the category whose objects are all Hilbert spaces

, and whose morphisms are linear maps between

spaces.
The
isomorphisms in

are all isometric isomorphisms.
Remark 0.1
The category of
Hilbert spaces has direct sums and is a Cartesian category.
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