International Journal of Mathematics and Mathematical Sciences
Volume 19 (1996), Issue 4, Pages 727-732
doi:10.1155/S0161171296001007
Strictly barrelled disks in inductive limits of quasi-(LB)-spaces
Carlos Bosch1
and Thomas E. Gilsdorf2
1Department of Mathematics I.T.A.M., Río Hondo #1, Col. Tizapán San Angel, D.F., México 01000, Mexico
2Department of Mathematics, University of North Dakota, Grand Forks 58202-8376, ND, USA
Abstract
A strictly barrelled disk B in a Hausdorff locally convex space E is a disk such that the linear span of B with the topology of the Minkowski functional of B is a strictly barrelled space. Valdivia's closed graph theorems are used to show that closed strictly barrelled disk in a quasi-(LB)-space is bounded. It is shown that a locally strictly barrelled quasi-(LB)-space is locally complete. Also, we show that a regular inductive limit of quasi-(LB)-spaces is locally complete if and only if each closed bounded disk is a strictly barrelled disk in one of the constituents.