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C(X) As A Dual Space

Published online by Cambridge University Press:  20 November 2018

E. G. Manes
E. G. Manes*
Affiliation:
Dalhousie University, Halifax, Nova Scotia
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It is known [1] that for compact Hausdorff X, C(X) is the dual of a Banach space if and only if X is hyperstonian, that is the closure of an open set in X is again open and the carriers of normal measures in C(X)* have dense union in X. With the desiratum of proving that C(X) is always the dual of some sort of space we broaden the concept of Banach space as follows. A Banach space may be comfortably regarded as a pair (E, B) where E is a topological linear space and B is a subset of E ; the requisite property is that the Minkowski functional of B be a complete norm whose topology coincides with that of E.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 3 , June 1972 , pp. 485 - 491
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Bade, W. G. et al., The space of all continuous functions on a compact Hausdorff space, Notes for Mathematics 2906, Section 8 (University of California at Berkeley, 1957).Google Scholar
2. Edelstein, M., On the representation of mappings of compact metrizable spaces as restrictions of linear transformations, Can. J. Math. 22 (1970), 372–375.Google Scholar
3. Mitchell, B., Theory of categories (Academic Press, New York, 1965).Google Scholar
4. Schaeffer, H. H., Topological vector spaces (Macmillan, New York, 1966).Google Scholar