Characterization of reflexivity by equivalent renorming (Q1827556)

Let \(X\) be a Banach space. Then \(X\) is reflexive if and only if it admits an equivalent { W2R} norm (introduced by \textit{V.\,D.\,Milman} [Russ. Math. Surv. 26, 79-163 (1972; Zbl 0238.46012)]). To obtain this main result, the authors establish the following statement: Let \(\Gamma\) be an~arbitrary set and let \(\| \cdot\| \) be Day's norm on \(\ell^\infty(\Gamma)\). Let \((x_n)\subset c_0(\Gamma)\) be such that \(\lim_{m,n\to\infty}2\| x_m\| ^2+ 2\| x_n\| ^2- \| x_m+x_n\| ^2=0\). Then \((x_n)\) has a weak cluster point~\(x\) if and only if \(\lim x_n=x\) (in the norm topology). In addition, the authors construct an example of an equivalent norm on~\(\ell^2\) which is W2R but not MLUR, and an equivalent norm on~\(\ell^2\) which is LUR but not W2R.