Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Proceedings of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46A05

Retrieve articles in all journals with MSC: 46A05

Additional Information

PII: S 0002-9939(1990)0994779-1
Keywords: Locally convex spaces, (strict) inductive limit, bounded set
Article copyright: © Copyright 1990 American Mathematical Society

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia