On dual Kadec norms

Abstract

Let $(X,\Vert \cdot \Vert )$ be a Banach space such that all $w^{⁎}$-convergent sequences in the dual unit sphere $S_{X^{⁎}}$ are also norm convergent. Then the weak^⁎ and norm topologies agree on $S_{X^{⁎}}$. By known results it implies that X has a renorming whose dual is locally uniformly rotund, hence also $C^1$-Fréchet smooth. In particular, X is an Asplund space. Our results also lend an alternative proof of the celebrated Josefson-Nissenzweig theorem.