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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  and
I. Tweddle
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
Information
Canadian Mathematical Bulletin , Volume 20 , Issue 2 , 01 June 1977 , pp. 271 - 272
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Bourbaki, N., Espaces Vectoriels Topologiques, Chapitres I et II, Hermann, Paris, 1966.Google Scholar
2. Day, M. M., Normed Linear Spaces, Springer-Verlag, Berlin, 1962.Google Scholar
3. Kelley, J. L., I. Namioka et al, Linear Topological Spaces, Van Nostrand, Princeton, 1963.Google Scholar
4. Köthe, G., Topological Vector Spaces I, Springer-Verlag, Berlin, 1969.Google Scholar
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