Projection methods in different Banach-like spaces

Abstract

Summary

In a uniformly convex Banach space, the existence and uniqueness of a nearest point from a point outside a given sub- set are guaranteed when the given subset is nonempty, closed and convex, which is so called the projection theorem.

Faried and El-Sharkawy [12] extended this fact to the com- plete countably normed space which is a linear space equipped with a countable number of pair-wise compatible norms. They required that the completion (of the space equipped with each one of all its norms) to be uniformly convex and the convex sub- set to be closed with respect to all norms.