Following the references to the accepted answer to Is the category of Banach spaces with contractions an algebraic theory? one discovers that there is an algebraic theory (infinitary) which is closely related to Banach-spaces-with-contractions but which contains some extra stuff (this wider category is called the category of *totally convex spaces*). This category has a remarkably nice presentation, and the relevant paper (MR1223636) also discusses Banach algebras (not algebraic) and $C^*$-algebras (algebraic).

Inspired by this, my question is:

- What's the nearest algebraic theory for inner product spaces?

There's also a connection to the question In what sense are fields an algebraic theory?. Fields aren't an algebraic theory, but there are "nearby" algebraic theories that can be used instead. Most often used is that of commutative, unital rings but there's also the theory of meadows which is "nearer" in some vague sense.

Also, in my answer to Clifford algebra as an adjunction?, I argued that vector+spaces+with+bilinear+form was sort of algebraic over *pointed* sets (I'd like an expert to check that if one's around!). But in this question, I don't want to change the underlying category (i.e. that of sets) but rather change (as little as possible) the category of inner product spaces.

Let me give a little more detail on the kind of answer I'd like to see. If one thinks about defining an inner product space, then it feels almost as though it is algebraic. The vector space structure is certainly algebraic and then on top of that there is a bilinear form with certain properties. All but one of these properties is expressed as an equality so these look like the usual sort of identities that one gets in an algebraic theory. However, there are two stumbling blocks:

- The positive definiteness of the inner product (and nondegeneracy, depending on whether you interpret "positive definite" to imply this or not)
- The codomain of the inner product.

The first is easy to deal with: simply allow arbitrary symmetric bilinear (or sesquilinear if over $\mathbb{C}$) forms. That's not a huge problem.

The second is more problematic. "Operations" in an algebraic theory go from products of the "base" object to the base object itself, $X^n \to X$, not to some other object no matter how special. One way around this is to start in the monoidal category $\operatorname{Vect}$ and use PROPs but that's not what I want. So we need to "beef up" the binary form to a genuine operation. One possibility would be to define a ternary operation $X^3 \to X$ which "morally" is the operation $(u,v,w) \mapsto \langle u, v \rangle w$.

Then, of course, there should be a variety (ha ha) of identities that this new operation should satisfy. So one set of possible answers to this question will be simply listing these identities. However, there may be other ways of making a bilinear form into an actual operation so other answers could give suggestions for these. It would also be interesting to see, for any particular answer, an example of a model of the theory that isn't a vector space with a symmetric bilinear form.