A note on the inheritance of properties of locally convex spaces by subspaces of countable codimension

Abstract

A locally convex space E is said to be c-barrelled if every countable weak* bounded subset of its topological dual E' is equicontinuous; to have property (C) if every weak* bounded stubset of E' is relatively weak* compact; to have property (S) if E' is weak* sequentially complete. If a locally convex space possesses any of the above properties, then so do all of its linear subspaces of countable codimension. Examples are furnished to show that the mentioned properties are distinct from each other. In [4], the authors proved that a linear subspace of countable codimension in a barrelled space is barrelled. In this paper, we see that the properties of a space being w-barrelled and having weak*sequentially complete dual are inherited by subspaces of countable codimension. Only a partial result is obtained concerning the seemingly most important property, that of being a Mackey space. The paper concludes with examples to demonstrate that the various concepts discussed are, in fact, disjoint. 1. The notation will be essentially that used by J. Horvath [3]. If (E, F) is a dual pairing (E and F not necessarily separating points), then o-(E, F) will denote the topology on E of pointwise convergence on F. The polar A' of a subset A of E is the set {If F: (a,f)l ? 1 for all aEA }. -r(F, E) will denote the topology on F of uniform convergence on o-(E, F)-compact subsets of E. The vector space of continuous linear functionals on a locally convex space E will be designated E'. A (not necessarily Hausdorff) locally convex space E is said to be barrelled if every closed, balanced, convex, absorbing subset of E is a neighborhood of 0 or, equivalently, if every o-(E', E)-bounded subset of E' is equicontinuous. A locally convex space E is said to be w-barrelled if every countable, o-(E', E)-bounded subset of E' is equicontinuous; to have property (C) if every o(E', E)-bounded subset of E' is relatively o-(E', E)-countably compact; to have property (S) if E' is o-(E', E)-sequentially complete; to be a Mackey space if it has the topology r(E, E'). The codimension of a linear subspace M of Presented to the Society, January 24, 1969; received by the editors June 10, 1970. AMAS 1970 subject classifications. Primary 46A05, 46A15.