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Transactions of the American Mathematical Society

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On Kōmura's closed-graph theorem


Author: Michael H. Powell
Journal: Trans. Amer. Math. Soc. 211 (1975), 391-426
MSC: Primary 46A30
MathSciNet review: 0380339
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Abstract: Let $ (\alpha )$ be a property of separated locally convex spaces. Call a locally convex space $ E[\mathcal{J}]$ an $ (\bar \alpha )$-space if $ \mathcal{J}$ is the final topology defined by $ {\{ {u_i}:{E_i}[{\mathcal{J}_i}] \to E\} _{i \in I}}$, where each $ {E_i}[{\mathcal{J}_i}]$ is an $ (\alpha )$-space. Then, for each locally convex space $ E[\mathcal{J}]$, there is a weakest $ (\bar \alpha )$-topology on E stronger that $ \mathcal{J}$, denoted $ {\mathcal{J}^{\bar \alpha }}$.

Kōmura's closed-graph theorem states that the following statements about a locally convex space $ E[\mathcal{J}]$ are equivalent:

(1) For every $ (\alpha )$-space F and every closed linear map $ u: F \to E[\mathcal{J}]$, u is continuous.

(2) For every separated locally convex topology $ {\mathcal{J}_0}$ on E, weaker than $ \mathcal{J}$, we have $ \mathcal{J} \subset \mathcal{J}_0^{\bar \alpha }$.

Much of this paper is devoted to amplifying Kōmura's theorem in special cases, some well-known, others not.

An entire class of special cases, generalizing Adasch's theory of infra-(s) spaces, is established by considering a certain class of functors, defined on the category of locally convex spaces, each functor yielding various notions of ``completeness'' in the dual space.


References [Enhancements On Off] (What's this?)

  • [1] Norbert Adasch, Tonnelierte Räume und zwei Sätze von Banach, Math. Ann. 186 (1970), 209–214 (German). MR 0467230
  • [2] J. Dacord and M. Jourlin, Sur les pré compacts d'un éspace localement convexe, C. R. Acad. Sci. Paris Sér. A-B 274 (1972), A463-A466. MR 47 #751.
  • [3] Marc De Wilde, Théorème du graphe fermé et espaces à réseau absorbant, Bull. Math. Soc. Sci. Math. R. S. Roumanie 11 (59) (1967), 225–238 (1968) (French). MR 0230103
  • [4] Marc De Wilde, Sur le théorème du graphe fermé, C. R. Acad. Sci. Paris Sér. A-B 265 (1967), A376–A379 (French). MR 0222597
  • [5] Marc De Wilde, Réseaux dans les espaces linéaires à semi-normes, Mém. Soc. Roy. Sci. Liège Coll. in-8^{∘} (5) 18 (1969), no. 2, 144 (French). MR 0285885
  • [6] V. Eberhardt, Durch Graphensätze definierte lokalkonvexe Räume, Dissertation, Munich, 1972.
  • [7] John Horváth, Topological vector spaces and distributions. Vol. I, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966. MR 0205028
  • [8] Gottfried Köthe, Topological vector spaces. I, Translated from the German by D. J. H. Garling. Die Grundlehren der mathematischen Wissenschaften, Band 159, Springer-Verlag New York Inc., New York, 1969. MR 0248498
  • [9] Yukio Kōmura, On linear topological spaces, Kumamoto J. Sci. Ser. A 5 (1962), 148–157 (1962). MR 0151817
  • [10] Vlastimil Pták, Completeness and the open mapping theorem, Bull. Soc. Math. France 86 (1958), 41–74. MR 0105606
  • [11] Manuel Valdivia Ure na, The general closed graph theorem in locally convex topological vector spaces, Rev. Acad. Ci. Madrid 62 (1968), 545–551 (Spanish, with English summary). MR 0240593

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DOI: https://doi.org/10.1090/S0002-9947-1975-0380339-9
Article copyright: © Copyright 1975 American Mathematical Society