Nearest point problem in countably normed spaces

TL;DR

This paper investigates the existence and uniqueness of common nearest points in countably normed spaces, establishing conditions under which such points exist and are unique, especially in uniformly convex completions.

Contribution

It proves the existence of common nearest points in countably normed spaces for compact subsets across all norms and establishes their uniqueness in uniformly convex completions.

Findings

Existence of common nearest points for compact sets in all norms.

Uniqueness of the common nearest point in uniformly convex completions.

Conditions under which nearest points are guaranteed in countably normed spaces.

Abstract

In a countably normed space which is a linear space equipped with a countable number of pair-wise compatible norms, we prove the existence of a common nearest point (in all norms) from a point outside a nonempty subset if this subset is compact with respect to all norms. We also prove the uniqueness of that common nearest point if the completion of the space equipped with only one of its norms is uniformly convex.