Dieudonné-Schwartz theorem in inductive limits of metrizable spaces. II

Qiu, Jing Hui

doi:10.1090/S0002-9939-1990-0994779-1

Dieudonné-Schwartz theorem in inductive limits of metrizable spaces. II
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by Jing Hui Qiu

Proc. Amer. Math. Soc. 108 (1990), 171-175

DOI: https://doi.org/10.1090/S0002-9939-1990-0994779-1

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Abstract:

The Dieudonné-Schwartz Theorem for bounded sets in strict inductive limits does not hold for general inductive limits $E = {\text {ind lim }}{E_n}$ . It does if all the ${E_n}$ are Fréchet spaces and for any $n \in N$ there is $m\left ( n \right ) \in N$ such that $\bar E_n^{{E_p}} \subset {E_{m\left ( n \right )}}$ for all $p \geq m\left ( n \right )$. A counterexample shows that this condition is not necessary. When $E$ is a strict inductive limit of metrizable spaces ${E_n}$ , this condition is equivalent to the condition that each bounded set in $E$ is contained and bounded in some $\left ( {{E_n},{\xi _n}} \right )$. Also, some interesting results for bounded sets in inductive limits of Fréchet spaces are given.

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