A note on Fréchet-Montel spaces

Lindström, Mikael

doi:10.1090/S0002-9939-1990-0994780-8

A note on Fréchet-Montel spaces
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Proc. Amer. Math. Soc. 108 (1990), 191-196 Request permission

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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