In this simple note http://arxiv.org/abs/0907.1813 (to appear in Colloq. Math.), Rossi and I proved a characterization in terms of "inversion of Riesz representation theorem".

Here is the result: let $X$ be a normed space and recall Birkhoff-James ortogonality: $x\in X$ is orthogonal to $y\in X$ iff for all scalars $\lambda$, one has $||x||\leq||x+\lambda y||$.

Let $H$ be a Hilbert space and $x\rightarrow f_x$ be the Riesz representation. Observe that $x\in Ker(f_x)^\perp$, which can be required using Birkhoff-James orthogonality:

**Theorem:** Let $X$ be a normed (resp. Banach) space and $x\rightarrow f_x$ be an isometric mapping from $X$ to $X^*$ such that

1) $f_x(y)=\overline{f_y(x)}$

2) $x\in Ker(f_x)^\perp$ (in the sense of Birkhoff and James)

Then $X$ is a pre-Hilbert (resp. Hilbert) space and the mapping $x\rightarrow f_x$ is the Riesz representation.

thinkI've seen a proof (in work of Daws) that if E and F are Banach spaces and we have a closed-range unital homomorphism from A(E) into B(F), then E is isomorphic to a weakly complemented subspace of F. If I have remembered this correctly, then it would answer the natural isomorphic variant of Philip's question. $\endgroup$ – Yemon Choi Mar 2 '11 at 10:14]. Namely, if the closed disk $\{\lambda : |\lambda| \leq \|T\|\}$ is a spectral set for every $T$ continuous, linear operator on a Banach space $X$ then $X$ is a Hilbert space (the converse was previously proven by von Neumann). [] C. Foias , Sur certains théorèmes de J. von Neumann concernant les ensembles spectraux. Acta Sci. Math. (Szeged) 18 (1957), pp. 15–20 $\endgroup$ – santker heboln Aug 22 '11 at 23:551more comments