TITLE: Is the category of Banach spaces with contractions an algebraic theory?
QUESTION [12 upvotes]: 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).

REPLY [7 votes]: 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.