The Theory of S-Spaces

Abstract

The theory of S-spaces is due to T. Husain [14]. Some of the important properties of S-spaces are the following: (a) they are not necessarily metrizable, although every metrizable 1.c. space is an S-space; (b) the Krein—Šmulian theorem which is true for Fréchet spaces can also be proved for complete S-spaces; (c)the completion of an S-space is B-complete; (d) every subspace of an S-space E is an S-space provided E satisfies a closure property (see the main text); (e) the dual E′^c of a complete S-space E is B _r -complete, provided E satisfies the closure property.