### Countable-codimensional subspaces of locally convex spaces

J. H. Webb
1973 Proceedings of the Edinburgh Mathematical Society
A barrel in a locally convex Hausdorff space E[x] is a closed absolutely convex absorbent set. A a-barrel is a barrel which is expressible as a countable intersection of closed absolutely convex neighbourhoods. A space is said to be barrelled {countably barrelled) if every barrel (a-barrel) is a neighbourhood, and quasi-barrelled (countably quasi-barrelled) if every bornivorous barrel (ff-barrel) is a neighbourhood. The study of countably barrelled and countably quasi-barrelled spaces was
more » ... ed spaces was initiated by Husain (2). It has recently been shown that a subspace of countable codimension of a barrelled space is barrelled ((4), (6)), and that a subspace of finite codimension of a quasi-barrelled space is quasi-barrelled (5). It is the object of this paper to show how these results may be extended to countably barrelled and countably quasi-barrelled spaces. It is known that these properties are not preserved under passage to arbitrary closed subspaces (3). Theorem 6 shows that a subspace of countable codimension of a countably barrelled space is countably barrelled. Let {£"} be an expanding sequence of subspaces of E whose union is E. 00 Then £' s f] E' n , and the reverse inclusion holds as well (1) if either (i) E'\o(E', E)~\ is sequentially complete or (ii) E'[fi(E', £)] is sequentially complete, and every bounded subset of E is contained in some E n . Theorem 1. Let E[%\ be a locally convex space with r = n(E, E') (the Mackey CO topology). Suppose E = (J E n , where {E n } is an expanding sequence of subspaces 00 of E. If E' = p) E' n , then E is the strict inductive limit of the sequence {£"}. I Proof. Let F[%] be any locally convex space, and T: E^F& linear mapping whose restriction T n to E n is continuous. We show that T is then continuous, which proves the result.