## Strictly barrelled disks in inductive limits of quasi-(LB)-spaces.(English)Zbl 0858.46005

Summary: 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 a 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.

### MSC:

 46A13 Spaces defined by inductive or projective limits (LB, LF, etc.) 46A08 Barrelled spaces, bornological spaces 46A30 Open mapping and closed graph theorems; completeness (including $$B$$-, $$B_r$$-completeness)
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