Reflexivity and approximate fixed points
Eva Matoušková, Simeon Reich · Studia Mathematica · 2003
A Banach space $X$ is reflexive if and only if every bounded sequence $\{x_n\}$ in $X$ contains a norm attaining subsequence. This means that it contains a subsequence $\{x_{n_k}\}$ for which $\sup_{f\in S_{X^*}}\limsup_{k\to \infty} f(x_{n_k})$ is attain