A variational principle in reflexive spaces with Kadec-Klee norm
نویسندگان
چکیده
We prove a variational principle in reflexive Banach spaces X with KadecKlee norm, which asserts that any Lipschitz (or any proper lower semicontinuous bounded from below extended real-valued) function in X can be perturbed with a parabola in such a way that the perturbed function attains its infimum (even more can be said — the infimum is well-posed). In addition, we have genericity of the points determining the parabolas. We prove also that the validity of such a principle actually characterizes the reflexive spaces with Kadec-Klee norm. This principle turns out to be an analytic counterpart of a result of K.-S. Lau on nearest points.
منابع مشابه
The fixed point property in Musielak-Orlicz sequence spaces
Abstract. In this paper, we give necessary and sufficient conditions for a point in a Musielak-Orlicz sequence space equipped with the Orlicz norm to be an H-point. We give necessary and sufficient conditions for a Musielak-Orlicz sequence space equipped with the Orlicz norm to have the Kadec-Klee property, the uniform Kadec-Klee property and to be nearly uniformly convex. We show that a Musiel...
متن کاملBanach Spaces Which Admit a Norm with the Uniform Kadec-klee Property
Several results are established about Banach spaces X which can be renormed to have the uniform Kadec-Klee property. It is proved that all such spaces have the complete continuity property. We show that the renorming property can be lifted from X to the Lebesgue-Bochner space L2(X) if and only if X is super-re exive. A basis characterization of the renorming property for dual Banach spaces is g...
متن کاملOn uniform Kadec-Klee properties and rotundity in generalized Cesàro sequence spaces
We consider the generalized Cesàro sequence spaces defined by Suantai (2003) and consider it equipped with the Amemiya norm. The main purpose of this paper is to show that ces (p) equipped with the Amemiya norm is rotund and has uniform Kadec-Klee property. 1. Introduction. In the whole paper, N and R stand for the sets of natural numbers and real numbers, respectively. Let (X, · ·) be a real n...
متن کاملOn geometrical properties of noncommutative modular function spaces
We introduce and study the noncommutative modular function spaces of measurable operators affiliated with a semifinite von Neumann algebra and show that they are complete with respect to their modular. We prove that these spaces satisfy the uniform Opial condition with respect to ρ̃-a.e.-convergence for both the Luxemburg norm and the Amemiya norm. Moreover, these spaces have the uniform Kadec–K...
متن کاملUniform Kadec-Klee Property in Banach Lattices
We prove that a Banach lattice X which does not contain the ln ∞uniformly has an equivalent norm which is uniformly Kadec-Klee for a natural topology τ on X. In case the Banach lattice is purely atomic, the topology τ is the coordinatewise convergence topology. 1980 Mathematics Subject Classification: Primary 46B03, 46B42.
متن کامل