×

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)
PDF BibTeX XML Cite
Full Text: DOI EuDML