On strong proximinality in normed linear spaces

Sahil Gupta,T D Narang

doi:10.17951/a.2016.70.1.19

Abstract

The paper deals with strong proximinality in normed linear spaces. It is proved that in a compactly locally uniformly rotund Banach space, proximinality, strong proximinality, weak approximative compactness and approximative compactness are all equivalent for closed convex sets. How strong proximinality can be transmitted to and from quotient spaces has also been discussed.

Highlights

Let W be a closed subset of a normed linear space (X, . )
The metric projection of X onto W is the set-valued map PW defined by PW (x) = {y ∈ W : x − y ≤ x − w f or all w ∈ W }
We prove some results concerning strong proximinality in normed linear spaces

Summary

Introduction

Let W be a closed subset of a normed linear space (X, . ). Let W be a closed subset of a normed linear space A proximinal subset W of a normed linear space A subset W of a normed linear space X is said to be strongly Chebyshev [1] for x ∈ X, if every minimizing sequence {yn} ⊆ W for x is convergent in W .

Results

Conclusion