A characterization of Hilbert spaces

The classical Jordan-Neumann paper is the first of a large literature on the subject on the characterization of the inner product space (Hilbert space) among the linear normed space X (Banach space). For each \theta ϵ R/2πZ, we define an operator R_{\theta} on space X + X. the characterization introduced in this paper has the following statement : the Banach space X is an Hilbert space if and only if R_theta turn out to be an isometry of X + X for some \theta . We remark that when \theta = π/4 we obtain the Jordan-Neumann classical result.