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.


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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