A variational characterization of reflexivity and strict convexity (Q2762831)

For a real Banach space \(X\) and \(f\in X^*\), \(f\neq 0,\) define \(F_f:X\to \mathbb R\) by \(F_f(x) = |x|^2 - 2f(x)/|f|\), \(x\in X.\)NEWLINENEWLINENEWLINEThe author proves the following variational characterization of reflexivity which reminds the classical James theorem: A real Banach space \(X\) is reflexive if an only if for every nonzero \(f\in X^*\) there exists \(u_f \in X\), \(|u_f|=1,\) which minimizes the quadratic functional \(F_f\). The existence and uniqueness of such an element \(u_f\) (for every \(f\in X^*\), \(f\neq 0\)) is equivalent to the reflexivity and strict convexity of the space \(X\).NEWLINENEWLINENEWLINENo applications of this result are given.