fa.functional analysis - When is a Banach space a Hilbert space?

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: $xin X$ is orthogonal to $yin X$ iff for all scalars $lambda$, one has $||x||leq||x+lambda y||$.

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

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

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

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

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