This is an announcement for the paper “Some remarks on smooth renormings of Banach spaces” by Petr Hájekhttps://arxiv.org/find/math/1/au:+Hajek_P/0/1/0/all/0/1, Tommaso Russohttps://arxiv.org/find/math/1/au:+Russo_T/0/1/0/all/0/1.

Abstract: We prove that in every separable Banach space X with a Schauder basis and a $C_k$-smooth norm it is possible to approximate, uniformly on bounded sets, every equivalent norm with a $C_k$-smooth one in a way that the approximation is improving as fast as we wish on the elements depending only on the tail of the Schauder basis. Our result solves a problem from the recent monograph of Guirao, Montesinos and Zizler..

The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.01384