fa.functional analysis - 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 this paper by Pelletier and Rosicky.

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.)