range of other situations, such as locally bifinite and quantale-valued relations, and positive $\ell_2$-matrices.

The problem
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The notion of Frobenius structure as defined above, which requires a unit, limits us to the *finite-dimensional case* in ${{\ensuremath{\mathbf{Hilb}}}}$, as the following lemma shows.

A Frobenius algebra in ${\ensuremath{\mathbf{Hilb}}}$ is unital if and only if it is finite-dimensional.

Sufficiency is shown in [@kock:frobenius 3.6.9]. Necessity follows from [@kaplansky:dualrings Corollary to Theorem 4].

In fact, a Frobenius structure on an object $A$ induces a *compact* (or *rigid*) structure on $A$, with $A$ as its own dual (see [@abramskycoecke:categoricalquantummechanics]). Indeed, put . In the category ${{\ensuremath{\mathbf{fHilb}}}}$, for example, $\eta
\colon {\ensuremath{\mathbb{C}}} \to {\ensuremath{\mathbb{C}}}^2 {\ensuremath{\otimes}}{\ensuremath{\mathbb{C}}}^2$ is an *entangled state preparation*: $$\eta_{\text{std}} 
  = \delta_{\text{std}}{\circ}\varepsilon^\dag_{\text{std}} 
  = (1 \mapsto \delta_{\text{std}}({{|} 0\rangle}+{{|} 1\rangle}))
  = (1 \mapsto {{|} 00\rangle} + {{|} 11\rangle}).$$ In general it is easy to see that $\eta$ indeed provides a dagger compact structure on $A$, with $A^*=A$: $${
 \begin{tikzpicture}[scale=0.33]  \PandocStartInclude{tikz/compact-1.tikz}\PandocEndInclude{input}{24}{23} \end{tikzpicture}}
  \; \stackrel{\text{(C)}}{=} \;
  {
 \begin{tikzpicture}[scale=0.5]  \PandocStartInclude{tikz/compact-2.tikz}\PandocEndInclude{input}{24}{23} \end{tikzpicture}}
  \; \stackrel{\text{(F)}}{=} \;
  {
 \begin{tikzpicture}[scale=0.5]  \PandocStartInclude{tikz/compact-3.tikz}\PandocEndInclude{input}{24}{23} \end{tikzpicture}}
  \; \stackrel{\text{(U)}}{=} \;
  {
 \begin{tikzpicture}[scale=0.5]  \PandocStartInclude{tikz/compact-4.tikz}\PandocEndInclude{input}{24}{23} \end{tikzpicture}}.$$ As is well-known, a compact structure exists only for finite-dimensional spaces in ${\ensuremath{\mathbf{Hilb}}}$. Thus to obtain a notion capable of being extended beyond the 