# Is there some way to see a Hilbert space as a C-enriched category?

The inner product of vectors in a Hilbert space has many properties in common with a hom functor. I know that one can make a projectivized Hilbert space into a metric space with the Fubini-Study metric and then follow Lawvere in considering it a $[0,\infty)$-enriched category; it's particularly nice because the distance between two vectors is a function of the inner product.

Is there a way of defining a monoidal structure on $\mathbb{C}$ such that the inner product is the hom so that we can think of a (perhaps projectivized) Hilbert space as $\mathbb{C}$-enriched?

-
Related question: mathoverflow.net/questions/476/… –  Loop Space Nov 29 '10 at 8:18
You might also be interested in John Baez's paper on the subject: arxiv.org/abs/q-alg/9609018 –  Qiaochu Yuan Nov 29 '10 at 9:23