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Weak Cauchy sequences in normed linear spaces

Published online by Cambridge University Press:  24 October 2008

D. J. H. Garling
D. J. H. Garling
Affiliation:
St John's College, Cambridge
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Extract

It follows from the Krein-Milman theorem that (c0) is not isomorphic to the dual of a Banach space. Using a technique due to Banach ((4), page 194) we shall extend this result to show that if a subspace of (c0) is isomorphic to the dual of a normed linear space, then it is finite dimensional (Proposition 1). Using this result, we shall show that if E is a normed linear space, the unit ball of which is contained in the closed absolutely convex cover of a weak Cauchy sequence, then Eis finite dimensional (Proposition 2). This result has applications to the Banach-Dieudonné theorem, and to the theory of two-norm spaces.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 60 , Issue 4 , October 1964 , pp. 817 - 819
Copyright
Copyright © Cambridge Philosophical Society 1964

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References

REFERENCES

(1)Alexiewicz, A. and Semadeni, Z.Linear functionals on two norm spaces. Studio Math. 17 (1958), 121–140.CrossRefGoogle Scholar
(2)Alexiewicz, A. and Semadeni, Z.The two-norm spaces and their conjugate spaces. Studia Math. 18 (1959), 275–293.CrossRefGoogle Scholar
(3)Alexiewicz, A. and Semadeni, Z.Some properties of two-norm spaces and a characterization of eflexivity for Banach spaces. Studia Math. 19 (1960), 115–132.CrossRefGoogle Scholar
(4)Banach, S.Théorie des Opérations Linéaires (Chelsea, New York, 1955).Google Scholar
(5)Garling, D. J. H.A generalized form of inductive-limit topology for vector spaces. Proc. London Math. Soc. (3), 14 (1964), 1–28.CrossRefGoogle Scholar
(6)Grothendieck, A.Espaces vectoriels topologiques (2nd edition; São Paulo, 1958).Google Scholar
(7)Krein, M. and Milman, D.On extreme points of regularly convex sets. Studia Math. 9 (1940), 133–138.CrossRefGoogle Scholar
(8)Wiweger, A.A topologisation of Saks spaces. Bull. Acad. Polon. Sci. Cl. III, 5 (1957), 773–777.Google Scholar
(9)Wiweger, A.Linear spaces with mixed topology. Studia Math. 20 (1961), 47–68.CrossRefGoogle Scholar