Norms That Generate the Same Wijsman Topology on Convex Sets
نویسنده
چکیده
Dedicated to R. T. Rockafellar on his 60th Birthday In a Banach space X, we introduce a criterion for comparing the Wijsman topologies that are induced by two equivalent norms of X on the hyperspace of closed convex sets C(X). Thereafter, we study the duality map associated with the unit ball of a given norm of X in relation to its composition with the polarity map. This more geometrical description of the norm allows us to give a direct proof of a known theorem (see 3] and 1]): If X is reeexive and the duality map is n-ton -usco, then the Wijsman topology coincides with the Mosco topology on C(X).
منابع مشابه
The Wijsman structure of a quantale-valued metric space
We define and study a quantale-valued Wijsman structure on the hyperspace of all non-empty closed sets of a quantale-valued metric space. We show its admissibility and that the metrical coreflection coincides with the quantale-valued Hausdorff metric and that, for a metric space, the topological coreflection coincides with the classical Wijsman topology. We further define an index of compactnes...
متن کاملWijsman Statistical Convergence of Double Sequences of Sets
In this paper, we study the concepts of Wijsman statistical convergence, Hausdorff statistical convergence and Wijsman statistical Cauchy double sequences of sets and investigate the relationship between them.
متن کاملM-FUZZIFYING TOPOLOGICAL CONVEX SPACES
The main purpose of this paper is to introduce the compatibility of $M$-fuzzifying topologies and $M$-fuzzifying convexities, define an $M$-fuzzifying topological convex space, and give a method to generate an $M$-fuzzifying topological convex space. Some characterizations of $M$-fuzzifying topological convex spaces are presented. Finally, the notion of $M$-fuzzifying weak topologies is obtaine...
متن کاملDifferential Forms Are Dual to a Differential Coalgebra
There is a fundamental asymmetry between algebras and their dual objects, coalgebras, namely that the dual of a coalgebra is an algebra, but the converse is only true in finite dimensions. We prove that there exists a differential graded coalgebra whose continuous dual is the differential graded algebra of differential forms. This coalgebra will be constructed as an explicit subspace of de Rham...
متن کاملA Continuity Theorem for Cores of Random Closed Sets
If a sequence of random closed sets Xn in a separable complete metric space converges in distribution in the Wijsman topology to X, then the corresponding sequence of cores (sets of probability measures dominated by the capacity functional of Xn) converges to the core of the capacity of X. Core convergence is achieved not only in the Wijsman topology, but even in the stronger Vietoris topology....
متن کامل