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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Dieudonné-Schwartz theorem in inductive limits of metrizable spaces. II
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by Jing Hui Qiu PDF
Proc. Amer. Math. Soc. 108 (1990), 171-175 Request permission

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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Additional Information
  • © Copyright 1990 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 108 (1990), 171-175
  • MSC: Primary 46A05
  • DOI: https://doi.org/10.1090/S0002-9939-1990-0994779-1
  • MathSciNet review: 994779