Bases and basic sequences in
Fspaces

Nigel J. Kalton and Joel H. Shapiro 
Studia Math. 56
(1976), 4761


Abstract: This paper is an extension of Kalton's work
on existence of basic sequences in Fspaces (not necessarily
locally convex) [Proc. Edinburgh Math. Soc. (2) 19 (1974/75),
no. 2, 151 167]. An Fspace E is said to have the restricted
HahnBanach extension property (RHBEP) if, for all closed
infinitedimensional subspaces L of E and for all nonzero x in
L, there exists a closed infinitedimensional subspace M of L
such that x lies in M. The following characterization of the
RHBEP is given: An Fspace E has the RHBEP if and only if every
closed infinitedimensional subspace contains a basic sequence.
Two new classes of Fspaces are introduced. An Fspace E is said
to be pseudoFr echet if the weak topology on each linear subspace
coincides on bounded sets with the weak topology of the whole
space. An Fspace E is said to be pseudoreflexive 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 Fspace to be pseudoFr echet or pseudoreflexive
in terms of shrinking and boundedly complete basic sequences.
This leads to the construction of nontrivial examples of nonlocally
convex pseudoFr echet spaces and pseudoreflexive spaces. 
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