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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A note on Fréchet-Montel spaces


Author: Mikael Lindström
Journal: Proc. Amer. Math. Soc. 108 (1990), 191-196
MSC: Primary 46A14; Secondary 46A06
DOI: https://doi.org/10.1090/S0002-9939-1990-0994780-8
MathSciNet review: 994780
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Abstract: Let $E$ be a Fréchet space and let ${C^b}\left ( E \right )$ denote the vector space of all bounded continuous functions on $E$. It is shown that the following statements are equivalent: (i) $E$ is Montel. (ii) Every bounded continuous function from $E$ into ${c_0}$ maps every absolutely convex closed bounded subset of $E$ into a relatively compact subset ${c_0}$. (iii) Every sequence in ${C^b}\left ( E \right )$ that converges to zero in the compact-open topology also converges uniformly to zero on absolutely convex closed bounded subsets of $E$.


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Article copyright: © Copyright 1990 American Mathematical Society