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Characterizations of ultrabarrelledness and barrelledness involving the singularities of families of convex mappings. (English) Zbl 0871.46001

Summary: The paper reveals that ultrabarrelled spaces (respectively barrelled spaces) can be characterized by means of the density of the so-called weak singularities of families consisting of continuous convex mappings that are defined on an open absolutely convex set and take values in a locally full ordered topological linear space (respectively locally full ordered locally convex space). The idea to establish such characterizations arose from the observation that, in virtue of well-known results, the density of the singularities of families of continuous linear mappings allows to characterize both the ultrabarrelled spaces and the barrelled spaces.

MSC:

46A08 Barrelled spaces, bornological spaces
46A40 Ordered topological linear spaces, vector lattices
26B25 Convexity of real functions of several variables, generalizations
52A41 Convex functions and convex programs in convex geometry
46A55 Convex sets in topological linear spaces; Choquet theory
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References:

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