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Proceedings of the American Mathematical Society

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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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DOI: https://doi.org/10.1090/S0002-9939-1990-0994780-8
Article copyright: © Copyright 1990 American Mathematical Society

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