International Journal of Mathematics and Mathematical Sciences
Volume 13 (1990), Issue 3, Pages 607-610
doi: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 Jan Kucera. 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 limit E=indlim En 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 indlim En 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.