On completeness in symmetric spaces

Abstract

In the literature completeness for symmetric spaces is done through the classical Cauchy criterion for metric spaces. However, unlike the situation in metric spaces a convergent sequence in a symmetric space is not necessarily a Cauchy sequence. In the paper we introduce a notion of convergence completeness for symmetric spaces and characterize completeness for these spaces without appealing to the notion of a Cauchy sequence. The new notion is equivalent to completeness when restricted to the class of metric spaces.

Keywords: Completeness, convergence complete, extension of spaces, metric space, symmetric distance space.

Quaestiones Mathematicae 30(2007), 13–20