Boundedness and surjectivity in Banach spaces

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.