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](http://ncatlab.org/nlab/show/commutative+algebraic+theory), 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*).



Is the category of Banach spaces with contractions an algebraic theory?
I think this is some kind of infinitary algebraic theory, but that it is *not* a monadic adjunction. That is, if you take the "closed unit ball functor" $B$ from ${\bf Ban}\\_1$ to ${\bf Set}$ and the "free Banach space functor" $L: {\bf Set} \to {\bf Ban}\_1$, then $L$ is left adjoint to $B$ but this adjunction is not monadic (IIRC, and I often don't).


See, for instance, the first few pages of Pelletier and Rosicky, *[On the equational theory of $C^\*$-algebras](https://doi.org/10.1007/BF01196099)*, Algebra Universalis **30** (1993) pp 275-28.


You say something about a closed structure on ${\bf Ban}\_1$, if I understand this right then this is symmetric monoidal with the tensor being the projective tensor product of Banach spaces. That seems to be well known but little-used, although IMHO having this kind of perspective takes some of the tedium/clutter out of certain computations/constructions in my corner of functional analysis.


I think the ball functor from (Hilbert spaces & contractions) to Set doesn't have a left adjoint, but that's more of a guess than an intuition. Certainly the `natural' attempt to build a left adjoint falls over.


As for putting a closed structure on Hilb .... well, the fact that the natural norm on B(H) is not Hilbertian suggests to me that this won't work. (Put another way, the natural Hilbertian tensor product would dualise to only considering Hilbert-Schmidt class maps between your Hilbert spaces, which in infinite dimensions rules out the identity morphism.)


