On operators which attain their norm at extreme points

Abstract.

We introduce a geometric property on Banach spaces, the E-property, that is implied by the $\lambda$-property and that implies the Bade property, although these properties are different. By mean of the E-property we characterize the topological dimension of compact metric spaces. If $(X_n)_{n\in {\Bbb N}}$ is a sequence of Banach spaces and $p \in [1,+ \infty)$ we relate the E-property of $(\oplus X_n) _p$ with the E-property of every X _n . Finally, if K is a compact Hausdorff space and X is a Banach space, we study the E-property on the dual space of C (K, X) .