Boundedness and surjectivity in Banach spaces

Abstract

We define the ($w^{\ast }$-) boundedness property and the ($w^{\ast }$-) surjectivity property for sets in normed spaces. We show that these properties are pairwise equivalent in complete normed spaces by characterizing them in terms of a category-like property called ($w^{\ast }$-) thickness. We give examples of interesting sets having or not having these properties. In particular, we prove that the tensor product of two $w^{\ast }$-thick sets in $\Xastast$ and $\Yast$ is a $w^{\ast }$-thick subset in $L(X,Y{)}^{\ast }$ and obtain as a concequense that the set $w^{\ast }-exp{B}_{K(l_2{)}^{\ast }}$ is $w^{\ast }$-thick.

15 pages