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 e-prints
- Pub Date:
- May 1996
- DOI:
- 10.48550/arXiv.math/9605213
- arXiv:
- arXiv:math/9605213
- Bibcode:
- 1996math......5213B
- Keywords:
-
- Mathematics - Functional Analysis;
- 46B20;
- 46B22