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Some forms of the closed graph theorem

Published online by Cambridge University Press:  24 October 2008

N. J. Kalton
N. J. Kalton
Affiliation:
Warwick University
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Extract

In this paper we shall establish some forms of the closed graph theorem for locally convex spaces, using the approach of Pták(17). Our interest is in classifying pairs of locally convex spaces (E, F) which have the property that every closed graph linear mapping T: E → F is continuous; if (E, F) has this property then we shall say that (E, F) is in the class ℒ if is a particular class of locally convex spaces then ℒ() is the class of all E such that (E, F)∈ℒ for all F ∈ .

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 70 , Issue 3 , November 1971 , pp. 401 - 408
Copyright
Copyright © Cambridge Philosophical Society 1971

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References

REFERENCES

(1)Bachelis, G. F. and Rosenthal, H. P. On unconditionally converging series and biorthogonal systems in a Banach space (to appear).Google Scholar
(2)Banach, S.Theorie des Operations Lineares (Warsaw, 1932).Google Scholar
(3)Bennett, G. and Kalton, N. J. On FK-spaces containing c 0, (to appear).Google Scholar
(4)Bourbaki, N.Elements de Mathématique, Livre 3, Topologie Generale (Paris, 1959).Google Scholar
(5)Bourbaki, N.Elements de Mathématique, Livre 6, Integration (Paris, 1959).Google Scholar
(6)Civin, P. and Yood, B.Quasi-reflexive spaces. Proc. Amer. Math. Soc. 8 (1957), 906–911.CrossRefGoogle Scholar
(7)Dunford, N. and Schwartz, J. T.Linear Operators, Vol. 1 (New York, 1955).Google Scholar
(8)Grothendieck, A.Sur les espaces (F) et (DF). Summa Brasil. Math. 3 (1954), 57–123.Google Scholar
(9)Isbell, J. R. Mazur's theorem. General Topology and its Relations to Morden Analysis and Algebra (Prague Symposium) (Prague, 1962).Google Scholar
(10)Kelley, J. L.General Topology (New York, 1955).Google Scholar
(11)Köthe, G.Topological Vector Spaces (Berlin, 1969).Google Scholar
(12)Mackey, G. W.On infinite dimensional linear spaces. Trans. Amer. Math. Soc. 57 (1945), 155–207.CrossRefGoogle Scholar
(13)Mahowald, M.Barrelled spaces and the closed graph theorem. J. London Math. Soc. 36 (1961), 108–110.CrossRefGoogle Scholar
(14)McIntosh, A. G.On the closed graph theorem. Proc. Amer. Math. Soc. 20 (1969), 397–404.CrossRefGoogle Scholar
(15)McWilliams, R. D.On certain Banaoh spaces which are ω*-sequentially dense in their duals. Duke Math. J. 37 (1970), 121–126.CrossRefGoogle Scholar
(16)Mrowka, S.On the form of pointwise continuous positive functionals and isomorphisms of function spaces. Studio. Math. 21 (1961), 1–14.CrossRefGoogle Scholar
(17)Pták, V.Completeness and the open mapping theorem. Bull. Soc. Math. France 86 (1958), 41–74.CrossRefGoogle Scholar
(18)Webb, J. H.Sequential convergence in locally convex spaces. Proc. Cambridge Philos. Soc. 64 (1968), 341–364.CrossRefGoogle Scholar
(19)Wilansky, A.Topics in Functional Analysis (Berlin, 1967).CrossRefGoogle Scholar