PROCEEDINGS OF THE AMERICAN MATHEMATICALSOCIETY Volume 53, Number 1, November 1975 A CLOSED GRAPH THEOREM S. O. IYAHEN ABSTRACT. A closed graph theorem is proved, implying that a Haus- dorff locally convex space E need not be barrelled if every closed linear map from £ into F is continuous, where F is a reflexive Frechet or LF- space or a space of distributions. A Hausdorff locally convex space (F, r) is barrelled if and only if for each Banach space (F, if), a closed linear map from (E, r) into (F, <f) is necessarily continuous. This result of M. Mahowald [2, Theorem 2.2] is shown here (Corollary) to be false if instead (F, <f) is a reflexive Frechet or LF-space. Theorem. Let (E, r) be a Hausdorff locally convex space. Suppose that for a family (S ) j, of absolutely convex compact subsets, there is no locally convex topology on E strictly finer than r which coincides with r on each Sa. Let (F, £) be a Hausdorff locally convex space in which every closed bounded set is compact. If every closed linear map from any Banach space into (F, <f) is bounded, then a closed linear map f from (E, t) into (F, cf ) is necessarily continuous. Proof. Let E„ denote the linear span of S„, and r the r-induced topology on E . Let r)a denote the Banach space topology on Ea with the sequence (Sa/w)°°_ , of sets as a base of neighborhoods of the origin. As the graph of the restriction map of / to Ea is closed in (Ea, ra) x (F, cf ), it is necessarily closed in (Ea, r)a) x (F, <f) since r]a is finer than r . By the hypothesis then, the closure Ta of f(S ) in (F, cf) is compact. The graph G of the restriction map fa of / to Sa is closed in the compact Hausdorff space Xa = Sa x Ta, where Sa, Ta are considered under the r- and cf-induced topologies, respectively. Let Ps , PT be the projection maps of X onto Sa, Ta respectively, a a and P-, the restriction of Pr to G„. As P~ is continuous and one-to- a a. a one, it is necessarily a homeomorphism, since G is compact. Therefore each map fa = P o P _ ,: S —> (F, <f) is continuous. This implies that a Ga f: (E, t) -> (F, rf) is continuous. In the Theorem, (F, t) could be the dual of any complete Hausdorff locally convex space under the topology of compact convergence [1, p. 16, Received by the editors September 11, 1974. AMS (MOS) subject classifications (1970). Primary 46A30; Secondary 46A07. Key words and phrases. Closed graph theorem, Montel and reflexive spaces. .nr Copyright © 1975. American Mathematical Society License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 166 S. O. IYAHEN Example B], and (F, £) could be any Montel Frechet or LF-space or any of the distribution spaces g, &', S, £>', 3", J',QM,0^, 0c, £V by [3]. Corollary. Let (E, r) be the dual of a complete Hausdorff locally convex space under the Mackey topology r. If (F, £) is a reflexive Frechet or LF- space or any of the distribution spaces above, then any closed linear map f from (E, t) into (F, £) is necessarily continuous. Proof. Let t. denote the topology on E of compact convergence and £. the weak topology on F associated with £. The graph of / is necessarily closed in (E, rjx (F, £Q), and by the Theorem then /: (E, ry> —>(F, £) is continuous. This implies that /: (E, r) —> (F, £) is continuous. REFERENCES 1. D. J. H. Garling, A generalized inductive limit topology for vector spaces, Proc. London Math. Soc. 14 (1964), 1-28. MR 28 #4330. 2. M. Mahowald, Barrelled spaces and the closed graph theorem, J. London Math. Soc. 36 (1961), 108-110. MR 23#A2728. 3. L. Schwartz, Sur le theoreme du graphe ferme, C. R. Acad. Sci. Paris Ser. A-B 263 (1966), A602-A605. MR 34 #6494. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF BENIN, BENIN CITY, NIGERIA License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use.