Legendre forms in reflexive Banach spaces (Q1624154)

A quadratic form \(Q\) from a normed space \(X\) in the reals is called a Legendre form if \(Q\) is sequentially weakly l.s.c.\ and if \(x_k\rightharpoonup x\) and \(Q(x_k)\rightarrow Q(x)\) imply \(x_k\rightarrow x\). The following statement is proved in detail. Theorem. If a reflexive Banach space possesses a Legendre form, then the space is Hilbertizable. This is not true for non-reflexive Banach spaces.