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Proceedings of the American Mathematical Society

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



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

Author: Jing Hui Qiu
Journal: Proc. Amer. Math. Soc. 108 (1990), 171-175
MSC: Primary 46A05
MathSciNet review: 994779
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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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Keywords: Locally convex spaces, (strict) inductive limit, bounded set
Article copyright: © Copyright 1990 American Mathematical Society