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Proceedings of the American Mathematical Society

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Dieudonné-Schwartz theorem in inductive limits of metrizable spaces


Author: Jing Hui Qiu
Journal: Proc. Amer. Math. Soc. 92 (1984), 255-257
MSC: Primary 46A05
DOI: https://doi.org/10.1090/S0002-9939-1984-0754714-5
MathSciNet review: 754714
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Abstract: The Dieudonné-Schwartz Theorem for bounded sets in strict inductive limits does not hold for general inductive limits $E = {\operatorname {ind}}\lim {{\text {E}}_{\text {n}}}$. It does if each $\bar E_n^E \subset {E_{m\left ( n \right )}}$ and all the ${E_n}$ are Fréchet spaces. 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 in some ${E_n}$.


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Keywords: Locally convex spaces, (strict) inductive limit, bounded set
Article copyright: © Copyright 1984 American Mathematical Society