On operators which attain their norm at extreme points

We introduce a geometric property on Banach spaces, the E-property, that is implied by the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ \lambda $\end{document}-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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ (X_n)_{n\in {\Bbb N}} $\end{document} is a sequence of Banach spaces and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ p \in [1,+ \infty) $\end{document} we relate the E-property of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ (\oplus X_n) _p $\end{document} with the E-property of every Xn. 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) .