Bases and basic sequences in
F-spaces
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Nigel J. Kalton and Joel H. Shapiro
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Studia Math. 56
(1976), 47--61
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| 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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