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(HM)-Spaces and Measurable Cardinals

Published online by Cambridge University Press:  20 November 2018

José A. Facenda Aguirre
José A. Facenda Aguirre*
Affiliation:
Facultad de Matemáticas Universidad de Sevilla C) Tarfia S/N. Sevilla (12), Spain
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Abstract

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A locally convex space E is called an (HM)-space if E has invariant nonstandard hulls. In this paper we prove that if E is an (HM)-space, then E is a T(μ)-space, where μ is the first measurable cardinal. This is equivalent to say that in an (HM)-space, with dim(E)≧μ, does not exist a continuous norm. With this result, we prove that there exists an inductive semi-reflexive space E such that the bounded sets in E are finite-dimensional but E is not an (HM)-space. Thus, we answer negatively to an open problem raised up by Bellenot. In this paper, we do not use nonstandard analysis.

Keywords

46A0503E55
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 27 , Issue 1 , 01 March 1984 , pp. 53 - 57
Copyright
Copyright © Canadian Mathematical Society 1984

References

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