${{\ensuremath{\mathbf{Mat}}}}({\ensuremath{\mathbb{C}}})$ is given by ${{{\ensuremath{\mathbf{Mat}}}}_{\ell^2}({\ensuremath{\mathbb{C}}})}$. This category has arbitrary sets as objects, and its morphisms $X \to Y$ are matrices such that for each $x \in X$, the family $\{ M(x, y) \}_{y \in Y}$ is $\ell^2$[-]{}summable; and for each $y \in Y$, the family $\{ M(x, y)
        \}_{x \in X}$ is $\ell^2$[-]{}summable. The category ${{\ensuremath{\mathbf{Hilb}}}}$ is equivalent to a (nonfull) subcategory of ${{{\ensuremath{\mathbf{Mat}}}}_{\ell^2}({\ensuremath{\mathbb{C}}})}$ [@blanketal:hilbert Theorem 3.1.7].

### Graphical Calculus {#graphical-calculus .unnumbered}

We briefly recall the graphical calculus for symmetric monoidal dagger categories [@selinger:graphicallanguages]. This can be seen as a two-dimensional version of *Dirac notation*, which allows equational reasoning to be performed graphically in a sound and complete fashion. A morphism $f \colon X \to Y$ is represented pictorially as ${
 \begin{tikzpicture}[scale=0.5,font=\scriptsize]  \PandocStartInclude{tikz/morphism.tikz}\PandocEndInclude{input}{24}{23} \end{tikzpicture}}$, the identity on $X$ simply becomes ${
 \begin{tikzpicture}[thick,scale=0.5,font=\scriptsize]  \PandocStartInclude{tikz/id.tikz}\PandocEndInclude{input}{24}{23} \end{tikzpicture}}$, and composition and tensor products appear as follows. $${
 \begin{tikzpicture}[scale=0.5]  \PandocStartInclude{tikz/composition-lhs.tikz}\PandocEndInclude{input}{24}{23} \end{tikzpicture}}
  \;=\;
  {
 \begin{tikzpicture}[scale=0.5]  \PandocStartInclude{tikz/composition-rhs.tikz}\PandocEndInclude{input}{24}{23} \end{tikzpicture}}
  \qquad\qquad
  {
 \begin{tikzpicture}[scale=0.5]  \PandocStartInclude{tikz/tensor-lhs.tikz}\PandocEndInclude{input}{24}{23} \end{tikzpicture}}
  \;=\;
  {
 \begin{tikzpicture}[scale=0.5]  \PandocStartInclude{tikz/tensor-rhs.tikz}\PandocEndInclude{input}{24}{23} \end{tikzpicture}}$$ The symmetry isomorphism $\sigma$ is drawn as $\vcenter{\hbox{{
 \begin{tikzpicture}[scale=0.25]  \PandocStartInclude{tikz/swap.tikz}\PandocEndInclude{input}{24}{23} \end{tikzpicture}}}}$. The dagger is represented graphically by a horizontal reflection.

Dagger Frobenius algebras
-------------------------

Frobenius algebras are a classic notion in mathematics [@nakayama:frobenius]. A particular form of such algebras 