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

Quasi-bounded sets

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.