Bounded Sets in Inductive Limits

J. Kucera, K. McKennon
1978 Proceedings of the American Mathematical Society  
The Dieudonne-Schwartz theorem for bounded sets in strict inductive limits does not hold for general inductive limits. A set B bounded in an inductive limit E = ind lim E" of locally convex spaces may not be contained in any E". If, however, each E" is closed in E, then B is contained in some E", but may not be bounded there. Let Ex c E2 c . . . be a sequence of locally convex spaces and E = ind lim En their inductive limit (with respect to the identity maps id: En -»
doi:10.2307/2043189 fatcat:6ftycry2jzcq3mmqq7bt4ejlji