Consider the category of Banach spaces with contractions as morphisms (weak, so $\|T\| \le 1$). Is this an algebraic theory?
I suspect that this is true. The "operations" will be weighted sums, where the sum of the weights is at most $1$. The "free Banach space" on a set $X$ should be $\ell^1(X)$. (Note that the "underlying set" functor sends a Banach space to its unit ball.) So, part one:
- Is this correct? If so, can anyone supply a reference (unfortunately, searching for "algebraic theory" and "Banach" doesn't turn up anything obvious).
- Has anything useful/unusual come out of this point of view?
- If this is correct, then the algebraic theory seems to be commutative, in which case it's a symmetric closed category. Has this angle been used?
- The norm can't be encoded as an operation, can it be categorically recovered?
Part two says: can we do this for Hilbert spaces?
Edit: Anyone even vaguely intrigued by this question should read the paper linked in Yemon's answer. In particular, it also answers a question that I was going to ask as a follow-up: what's the nearest algebraic theory to Banach spaces (the answer being totally convex spaces).