Compactly generated space

Abstract: We introduce the concept of compactly lipschitzian functions taking values in a topological vector space F. We show that if F is finite dimensional the Lipschitz functions are compactly lipschitizian. We define the notions of generalized directional derivatives and subdifferentials for such functionsf taking values in an ordered topological vector space. It is shown that this notion of subdifferential coincides with the one of F. H. Clarke whenf is Lispchits and F=ℝ. Some formulas for this subdifferential concerning the cases of finite sum, composition, pointwise supremum and continuous sum are studied.

Abstract: The purpose of this paper is to show how the method, developed by M. Valdivia, of constructing projections in a weakly compactly generated Banach space X can be used to give a direct proof of the Lindelof property for the weak topology in X. Since this method extends to large classes of spaces we prove a Lindelof property for Banach spaces with a Valdivia compact weak* dual unit ball. Our result includes previous ones of K. Alster, R. Pol and S. P. Gul'ko for the space C(K) with the topology of pointwise convergence, where K is a Corson compact space. If X is a Banach space then the dual X* has the Radon-Nikodym property if and only if it is Lindelof in the topology of uniform convergence on the separable bounded subsets of X. Finally, if a dual space X* is weakly Lindelof then the bidual unit ball must be a Corson compact space in the weak* topology, so the product X* ×X* is weakly Lindelof, too. This result gives a positive answer to a problem of Corson in the case of dual spaces.