Strictly barrelled disks in inductive limits of quasi‐(LB)‐spaces

A strictly barrelled disk B in a Hausdorff locally convex space E isa disk such that the linear span of B with the topology of the Minkowskifunctional of B is a strictly barrelled space. Valdivia's closed graph theoremsare used to show that closed strictly barrelled disk in a quasi-(LB)-space isbounded. It is shown that a locally strictly barrelled quasi-(LB)-space islocally 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 strictlybarrelled disk in one of the constituents