A characterization of complete metric spaces

Abstract

A general formulation of the completeness argument used in the Bishop-Phelps Theorem and many other places has been given by Ekeland. It is shown that Ekeland's formulation characterizes complete metric spaces. A central idea in the proof of the Bishop-Phelps Theorem is the use of norm completeness and a partial ordering to produce a point where a linear functional attains its supremum on a closed bounded convex set. In fact, this completeness technique is useful in many situations as has been described in the surveys of Phelps [7] and Brezis-Browder [1]. Recently Ekeland [3] has given a very general formulation of this technique and has applied it to a wide variety of problems [4]. In the present note we show that Ekeland's formulation is actually equivalent to completeness for metric spaces. Ekeland's Theorem may be stated as follows: THEOREM 1. Let (M, d) be a complete metric space, and F: M -R U { + oo} a lower semicontinuous function, F 5 + x, bounded from below. Let > 0 be given and a point u & M such that