On weakly locally uniformly rotund Banach spaces (Q1288248)

Let \((E,\|\cdot\|)\) denote a normed space and \(S_E\) the unit sphere. The norm \(\|\cdot\|\) on a normed space \(E\) is said to be: locally uniformly rotund (LUR) if \(\lim_n\| x_n- x\|= 0\) whenever \(x_n\), \(x\in S_E\), \(n\in\mathbb{N}\), are such that \(\lim_n\| x_n+ x\|= 2\); weakly locally uniformly rotund (WLUR) if weak-\(\lim_n(x_n- x)= 0\) whenever \(x_n\), \(x\in S_E\), \(n\in\mathbb{N}\), are such that \(\lim_n\| x_n+ x\|= 2\). The main result of the present paper is the following: Let \(E\) be a normed space with a WLUR norm. Then \(E\) has an equivalent LUR norm.