English

On nicely smooth Banach spaces

Functional Analysis 2016-09-06 v1

Abstract

In this work, we obtain some necessary and some sufficient conditions for a space to be nicely smooth, and show that they are equivalent for separable or Asplund spaces. We obtain a sufficient condition for the Ball Generated Property (BGP), and conclude that Property (II)(II) implies the BGP, which, in turn, implies the space is nicely smooth. We show that the class of nicely smooth spaces is stable under coc_o and p\ell_p sums and also under finite 1\ell_1 sums; that being nicely smooth is not a three space property; and that the Bochner LpL_p spaces are nicely smooth if and only if XX is both nicely smooth and Asplund. A striking result obtained is that every equivalent renorming of a space is nicely smooth if and only if it is reflexive.

Keywords

Cite

@article{arxiv.math/9605213,
  title  = {On nicely smooth Banach spaces},
  author = {Pradipta Bandyopadhyay and Sudeshna Basu},
  journal= {arXiv preprint arXiv:math/9605213},
  year   = {2016}
}