Approximative Compactness and Radon-Nikodym Property inw∗Nearly Dentable Banach Spaces and Applications

Shaoqiang Shang,Yunan Cui

doi:10.1155/2015/921456

Abstract

Authors definew∗nearly dentable Banach space. Authors study Radon-Nikodym property, approximative compactness and continuity metric projector operator inw∗nearly dentable space. Moreover, authors obtain some examples ofw∗nearly dentable space in Orlicz function spaces. Finally, by the method of geometry of Banach spaces, authors give important applications ofw∗nearly dentability in generalized inverse theory of Banach space.

Highlights

Let (X, ‖ ⋅ ‖) be a real Banach space
Let N, R, and R+ denote the set of natural numbers, reals, and nonnegative reals, respectively
Since cow∗ (B(X∗) \ UA(x)) is a weak∗ compact set, by the separation theorem of locally convex space, there exists x ∈ S(X) such that inf {x (x∗) : x∗ ∈ A (x)} > sup {x (x∗) : x∗ ∈ cow∗ (B (X∗) \ UA(x))} . (6)

Summary

Introduction

Let (X, ‖ ⋅ ‖) be a real Banach space. S(X) and B(X) denote the unit sphere and the unit ball, respectively. Since cow∗ (B(X∗) \ UA(x)) is a weak∗ compact set, by the separation theorem of locally convex space, there exists x ∈ S(X) such that inf {x (x∗) : x∗ ∈ A (x)} > sup {x (x∗) : x∗ ∈ cow∗ (B (X∗) \ UA(x))} . 3. w∗ Nearly Dentability and Approximative Compactness and Continuity of Metric Projector Operator in Banach Spaces

Results

Conclusion