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On the Dimension of a Complete Metrizable Topological Vector Space

Published online by Cambridge University Press:  20 November 2018

J. O. Popoola
Affiliation:
University of Stirling, StirlingFK9 4LA Scotland
I. Tweddle
Affiliation:
University of Lagos, LagosNigeria
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The purpose of this note is to prove a result which is known to hold for Fréchet spaces [1, Chapitre II, §5, Exercise 24]. M. M. Day [2, p. 37] attributes the Banach space case to H. Löwig, although the earliest version that we have been able to find is that given by G. W. Mackey in [7, Theorem 1-1]. Recently H. E. Lacey has given an elegant proof for Banach spaces [5]. It is perhaps interesting to note that the non-locally convex case can be deduced from these known results which are established by duality arguments.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Bourbaki, N., Espaces Vectoriels Topologiques, Chapitres I et II, Hermann, Paris, 1966.Google Scholar
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5. Lacey, H. E., The Hamel dimension of any infinite dimensional separable Banach space is c, Amer.Math. Monthly 80 (1973), 298.Google Scholar
6. Löwig, H., Über die Dimension linearer Räume, Studia Math. 5 (1935), 18-23.Google Scholar
7. Mackey, G. W., On infinite-dimensional linear spaces, Trans. Amer. Math. Soc. 57 (1945), 155-207.Google Scholar