Mackey duals and almost shrinking bases

Abstract

Suppose (en) is a basis of a Banach space E, and that (e′n) is the dual sequence in E′. Then if (e′n) is a basis of E′ in the norm topology (i.e. (en) is shrinking) it follows that E′ is norm separable: it is easy to give examples of spaces E for which this is not so. Therefore there are plenty of spaces which cannot have a shrinking basis. This leads one to consider whether it might not be reasonable to replace the norm topology on E′ by one which is always separable (provided E is separable). Of course, the weak*-topology σ(E′, E) is one possibility (Köthe (17), p. 259); then it is trivial that (e′n) is a weak*-basis of E′. However, if the weak*-topology is separable, then so is the Mackey topology τ(E′, E) on E′, and so we may ask whether (e′n) is a basis of (E′,τ(E′, E)).