  We use non-standard analysis to define a category
$^\star\!\operatorname{Hilb}$ suitable for categorical quantum mechanics in
arbitrary separable Hilbert spaces, and we show that standard bounded operators
can be suitably embedded in it. We show the existence of unital special
commutative $\dagger$-Frobenius algebras, and we conclude
$^\star\!\operatorname{Hilb}$ to be compact closed, with partial traces and a
Hilbert-Schmidt inner product on morphisms. We exemplify our techniques on the
textbook case of 1-dimensional wavefunctions with periodic boundary conditions:
we show the momentum and position observables to be well defined, and to give
rise to a strongly complementary pair of unital commutative $\dagger$-Frobenius
algebras.
