Bases and basic sequences in F-spaces

Nigel J. Kalton and Joel H. Shapiro


Studia Math. 56 (1976), 47--61
Abstract: This paper is an extension of Kalton's work on existence of basic sequences in F-spaces (not necessarily locally convex) [Proc. Edinburgh Math. Soc. (2) 19 (1974/75), no. 2, 151 167]. An F-space E is said to have the restricted Hahn-Banach extension property (RHBEP) if, for all closed infinite-dimensional subspaces L of E and for all nonzero x in L, there exists a closed infinite-dimensional subspace M of L such that x lies in M. The following characterization of the RHBEP is given: An F-space E has the RHBEP if and only if every closed infinite-dimensional subspace contains a basic sequence. Two new classes of F-spaces are introduced. An F-space E is said to be pseudo-Fr echet if the weak topology on each linear subspace coincides on bounded sets with the weak topology of the whole space. An F-space E is said to be pseudo-reflexive if the weak topology is Hausdorff and if every bounded set is relatively compact in the weak topology of its closed linear span. We give criteria for an F-space to be pseudo-Fr echet or pseudo-reflexive in terms of shrinking and boundedly complete basic sequences. This leads to the construction of non-trivial examples of non-locally convex pseudo-Fr echet spaces and pseudo-reflexive spaces.


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