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, 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.)

One of the major-league experts on this topic is Michael Barr. In a message to the categories list (dated October 22, 2003), he writes:

"For Banach spaces, if you take as underlying functor the closed unit ball, it has an adjoint. It is not tripleable, however, but $C^*$-algebras are (with the unit ball underlying functor)."

"Tripleable" means the same as monadic. See the discussion in the book by Barr-Wells, Toposes, Theories, and Triples, section 4.4. I think that I saw somewhere though that the forgetful functor from Banach spaces and linear contractions to metric spaces with basepoint and contractions is monadic.

You may also be interested in the discussion on p59-61 of Makkai-Pare's "Accessible categories," which shows that Banach spaces and Hilbert spaces are both accessible categories by giving an explicit axiomatization in terms of relations and operations. They claim that Banach spaces are actually locally presentable, although their proof doesn't show this; it could also be deduced from (co)completeness since any complete or cocomplete accessible category is locally presentable. Local presentability is not the same as being algebraic, of course. Hilbert spaces, on the other hand, are not locally presentable (and hence not complete or cocomplete, and thus certainly not algebraic), since their category is self-dual and no locally presentable category (other than a poset) has a locally presentable dual.

In "Basic concepts of enriched category theory" Kelly mentions the category of Banach spaces as one of his examples of a symmetric monoidal closed category one can enrich over. I don't know whether there are interesting applications.