Weak Baire measurability of the balls in a Banach space

José Rodríguez¹

¹ Departamento de Análisis Matemático Universidad de Valencia Avda. Doctor Moliner 50 46100 Burjassot, Valencia, Spain and Instituto Universitario de Matemática Pura y Aplicada Universidad Politécnica de Valencia Camino de Vera s//n 46022 Valencia. Spain

Studia Mathematica, Tome 185 (2008) no. 2, pp. 169-176

Résumé

Let $X$ be a Banach space. The property $(\star)$ “the unit ball of $X$ belongs to ${\rm Baire}(X,{\rm weak})$” holds whenever the unit ball of $X^{*}$ is weak$^{*}$-separable; on the other hand, it is also known that the validity of $(\star)$ ensures that $X^{*}$ is weak$^{*}$-separable. In this paper we use suitable renormings of $\ell^{\infty}(\mathbb{N})$ and the Johnson–Lindenstrauss spaces to show that $(\star)$ lies strictly between the weak$^{*}$-separability of $X^{*}$ and that of its unit ball. As an application, we provide a negative answer to a question raised by K. Musia/l.

Détail

DOI : 10.4064/sm185-2-5

Keywords: banach space property star nbsp unit ball nbsp belongs baire weak holds whenever unit ball nbsp * weak * separable other known validity star ensures * weak * separable paper suitable renormings ell infty mathbb johnson lindenstrauss spaces star lies strictly between weak * separability nbsp * its unit ball application provide negative answer question raised nbsp musia