Countable-Codimensional Subspaces ofc0-Barrelled Spaces

Abstract

A HAUSDORFF locally convex space is said to be c,-barrelled (respectively cu-barrelled) if each sequence in the dual space t h a t converges weakly to 0 (res!,r:ctively t h a t is weakly ?.~oundecl), is equicontinuous. It is proved that if a c,,-barrelled space E has dual E' weakly sequentially complete, then every subsi'ace of countable codimensjon of E is c,-barrcllecl. %Ire prove that the hypothesis on E' cannot be dropped and we supply a n esniiiple of a complete c,,-hnrrellecl space with dual wealily sequentially coin1)lete that is not co-barrelled. Introdnetion and PreliriiinnriesFor the notations and terminology as well as for the not proved results, we will refer throughout t o [ l], [4] a i d [3].A HAVSDORFP locally convex space [E, t] or simply E with topological ctual E' is said t o be c,-barrelled (resp. co-barrelled) if each sequence t h a t converges t o 0 (resp. t h a t is bounded) in [E', a@', E ) ] is equicontinuous. E has the property (S) if [E', a@', E ) ] is sequentially complete. It is known [&] t h a t every subspace of countable coclimension of a w-barrelled space is w-barrelled. The proof of this inheritance property relies heavily on the sequential coinpleteness of [E', o(E', E ) ] . However, if E is c,,-barrelled, E does not posses necessarily the property (8) and E might contain (and in fact does) countable-codimensional subspaces that are not c,,-barrelled. I n the main result we prove that every countable-codimensional subspzce of a co-barrelled space with property (8) is c,-barrelled and we furnish an example showing t h a t the property (S) cannot be dropped.As usual, we will denote by p(E, E') the MACKEY topology associated to the chal pair {E, B'). We denote by to(&', E') the locally convex topology on E of the zcniform convergence o n all the a(E', E)-convergent to 0 sequences of E'. Obviously a locally convex space [ E , z] with dual E is c,-barrelled iff to(E, E') SZ. Moreover, for the ;ILACKEY topology we have 1) The author was supported by a Fulbright-MEC fellowship. The hospitality of the Department of Mathematics, University of RIaryland, is gratefully acknowledged and the author thanks Dr. J. H O R V~T H for his helpful comments. Current address: Departamento Matemiticas, Facultad cle Ciencias. 06071-Badajoz, Spain.