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%%% Primary Title: category of Hilbert spaces
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%%% Filename: CategoryOfHilbertSpaces.tex
%%% Version: 1
%%% Owner: bci1
%%% Author(s): bci1
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\begin{document}
\begin{definition}
The \emph{\htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} $\mathcal{H}ilb_f$ of finite-dimensional \htmladdnormallink{Hilbert spaces}}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} is defined as the category whose \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} are all finite-dimensional Hilbert spaces $\mathcal{H}_f$, and whose \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} are linear maps between $\mathcal{H}_f$ spaces.
The \emph{\htmladdnormallink{isomorphisms}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html}} in $\mathcal{H}ilb_f$ are all isometric isomorphisms.
\end{definition}
Furthermore, one also has the following, \htmladdnormallink{general definition}{http://planetphysics.us/encyclopedia/PreciseIdea.html} for any Hilbert space.
\begin{definition}
The \emph{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 \emph{isomorphisms} in $\mathcal{H}ilb$ are all isometric isomorphisms.
\end{definition}
\begin{remark}
The category of $\mathcal{H}ilb$ Hilbert spaces has direct sums and is a Cartesian category.
\end{remark}
\end{document}
