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International Journal of Mathematics and Mathematical Sciences
Volume 13 (1990), Issue 3, Pages 607-610
http://dx.doi.org/10.1155/S0161171290000849
Research notes

Quasi-bounded sets

Department of Mathematics, Washington State University, Pullman 99164-2930, Washington, USA

Received 5 May 1989; Revised 3 April 1990

Copyright © 1990 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

It is proved in [1] & [2] that a set bounded in an inductive limitE=indlimEn of Fréchet spaces is also bounded in some En iff E is fast complete. In the case of arbitrary locally convex spaces En every bounded set in a fast complete indlimEn is quasi-bounded in some En, though it may not be bounded or even contained in any En. Every bounded set is quasi-bounded. In a Fréchet space every quasi-bounded set is also bounded.