On nicely smooth Banach spaces
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 $(II)$ implies the BGP, which, in turn, implies the space is nicely smooth. We show that the class of nicely smooth spaces is stable under $c_o$ and $\ell_p$ sums and also under finite $\ell_1$ sums; that being nicely smooth is not a three space property; and that the Bochner $L_p$ spaces are nicely smooth if and only if $X$ 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 1996
 arXiv:
 arXiv:math/9605213
 Bibcode:
 1996math......5213B
 Keywords:

 Mathematics  Functional Analysis;
 46B20;
 46B22