Is the parallelogram law the only way to characterise norms induced by inner products?

Question

A norm can be induced by an inner product iff the parallelogram identity is satisfied. This is well known and has already been discussed on the site multiple times.

Are there other ways to characterise norms induced by inner products, or equivalently, other conditions that are equivalent to the parallelogram identity?

One could of course always modify trivially the parallelogram identity to obtain a "different" condition, but I'm asking about characterisations that are not "trivially equivalent" to it.

Answer 1

There is a fair-sized literature on this topic. Not that I have read much of it, but I do have this list of references:

• Dan Amir, Characterizations of Inner Product Spaces, Birkhäuser (1986).
• Neil Falkner, "A Characterisation of Inner Product Spaces", Amer. Math. Month. 100, 3 (Mar 1993), pp. 246-249.
• Desmond Fearnley-Sander & J. S. V. Symons, "Appollonius and Inner Products", Amer. Math. Month. 81, 9 (Nov 1974), pp. 990-993.
• E. R. Lorch, "On Certain Implications Which Characterize Hilbert Space", Ann. Math. (2) 49, 3 (Jul 1948), pp.523-532.
• Frederick A. Ficken, "Note on the Existence of Scalar Products in Normed Linear Spaces", Ann. Math. (2) 45, 2 (Apr 1944), pp.362-366.
• P. Jordan & J. v. Neumann, "On Inner Products in Linear, Metric Spaces", Ann. Math. (2) 36, 3 (Jul 1935), pp.719-723.

Starting from these, the references therein, and citations on e.g. Google Scholar, should provide more.

---

From Amir (1986), p.2f.:

Altogether the area has been active in the last fifty years (cf. the chronological references list) and there are, by now, hundreds of such characterizations scattered in more than 150 papers, many of them not easily available. Almost everyone who confronted such a characterization problem and managed to solve it, discovered later that the problem had been solved before or that the proof could have been considerably simplified applying former characterizations. The idea to write this survey occurred to me after such an experience, only, because of my ignorance, I underestimated the extent of such a work by an order of magnitude. Not only that new characterizations keep pouring in, but old ones keep popping out in unexpected places. Therefore I do not claim that this survey is comprehensive, notwithstanding its very restricted topic. It is concerned only with characterizations of inner product spaces by normed-space geometry and approximation theoretic properties. It concerns only the real case. This suffices for most purposes, since if $E$ is a normed space over the complex field which, as a space over the reals, has the inner product $\left\langle x, y \right\rangle_\mathbb{R}$, then $\left\langle x, y \right\rangle = \left\langle x, y \right\rangle_\mathbb{R} + i\left\langle x, iy \right\rangle_\mathbb{R}$ is an inner product for $E$ over the complex field (observe that $2\|x\|^2 = \|(1+i)x)\|^2 = 2\|x\|^2 + 2\left\langle x, ix \right\rangle_\mathbb{R}$, so that $\left\langle x, ix \right\rangle_\mathbb{R} = 0$).

[$\ldots$] The survey consists of about 350 numbered statements, each equivalent to the space $E$ being an inner product or Hilbert space, and the proofs of these equivalences. [$\ldots$]

Even this admittedly incomplete survey, published 33 years ago, has a bibliography of 155 items.