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Quasi-LB-spaces. (English) Zbl 0625.46006

In this article, the author continues his investigations on the closed graph theorem. He introduces the class of quasi-LB-spaces, which contains the LF-spaces and their strong duals and has good inheritance properties. A quasi-LB-representation in a locally convex (l.c.) space E is an ordered covering \(\{A_{\alpha};\alpha \in {\mathbb{N}}^{{\mathbb{N}}}\}\) of E by Banach discs. E is said to be a quasi-LB-space if it admits a quasi- LB-representation. Here are some results on these spaces: E is a quasi- LB-space if and only if it has an ordered strict web. A locally complete space E with a completing web is a quasi-LB-space. It follows that any locally complete webbed space has a strict web, which answers a question of de Wilde.
The author proceeds to present a theory which is based on quasi-LB- structures rather than on de Wilde’s strict webs. He demonstrates that quasi-LB-spaces and quasi-LB-representations are very useful to derive closed graph theorems, lifting theorems (for null and Cauchy sequences as well as precompact subsets) and localization properties. His methods (which are, in part, based on an extension of the classical ideas of Banach) are quite powerful and provide substantial simplifications.
In connection with closed graph theorems for quasi-LB-spaces, the new class of strictly barrelled spaces (which contains all totally barrelled spaces) is defined and studied. Any linear mapping, with closed graph, from a strictly barrelled space into a quasi-LB-space must be continuous. An inductive limit of a family of strictly barrelled spaces which is a quasi-LB-space must be ultrabornological. The space \(L_ b(E,F)\) of all continuous linear mappings from E into F, endowed with the topology of uniform convergence on the bounded subsets of E, is a quasi-LB-space if E is a countable inductive limit of metrizable and strictly barrelled spaces and F is a quasi-LB-space or if E is a countable inductive limit of metrizable spaces and F a locally complete quasi-LB-space.
Reviewer: K.D.Bierstedt

MSC:

46A30 Open mapping and closed graph theorems; completeness (including \(B\)-, \(B_r\)-completeness)
46A08 Barrelled spaces, bornological spaces
46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
46A04 Locally convex Fréchet spaces and (DF)-spaces
46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators
46A32 Spaces of linear operators; topological tensor products; approximation properties
46A45 Sequence spaces (including Köthe sequence spaces)
46A50 Compactness in topological linear spaces; angelic spaces, etc.
46M05 Tensor products in functional analysis
46M40 Inductive and projective limits in functional analysis
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