Ekeland's theorem and UC spaces

Abstract

As is well-known, completeness of a metric space is necessary and sufficient for the Ekeland variational principle to hold for lower bounded proper lower semicontinuous functions [F. Sullivan, A characterization of complete metric spaces, Proc. Am. Math. Soc. 83 (1981), pp. 345–346]. In this note we characterize the curious class of UC spaces, which sits properly between complete metric spaces and compact metric spaces, by a stronger variational principle of the Ekeland type. We also characterize these spaces in terms of a fixed point property of continuous maps