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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
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

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