Compact operators

Abstract

Corresponding to the strong (s) and weak (w) topologies for a Hilbert space H, there are four possible interpretations of continuity for a transformation from H into H: they are the ones suggested by the symbols (s → s), (w → w), (s → w), and (w → s). Thus, to say that A is continuous (s → w) means that the inverse image under A of each w-open set is s-open; equivalently it means that the direct image under A of a net s-convergent to f is a net w-convergent to Af. Four different kinds of continuity would be too much of a good thing; it is fortunate that three of them collapse into one.