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Countability, completeness and the closed graph theorem. (English) Zbl 0744.46003
Generalized functions, convergence structures, and their applications, Proc. Int. Conf., Dubrovnik/Yugosl. 1987, 375-381 (1988).

[For the entire collection see Zbl 0708.00008.]

The webs of M. de Wilde have made an enormous contribution to the closed graph theorems in locally convex spaces (cs). Although webs have a very intricate layered construction, two properties in particular have contributed to the closed graph theorem. First of all, webs possess a strong countability condition in the range space which suitably matches the Baire property of Fréchet spaces in the domain space; as a result the zero neighbourhood filter is mapped to a p-Cauchy filter, a filter attempting to settle down. Secondly webs provide a completeness condition which allow p-Cauchy filters to converge.

In [the first author, Sigma Ser. Pure Math. 5, 29-45 (1984; Zbl 0574.46003) and Math. Nachr. 116, 159-164 (1984; Zbl 0574.46004)] webs were examined in the context of convergence spaces. The webs gave rise to a convergence vector space (cvs), the webspace, and the countability and completeness properties of the web were reflected in similar properties of the web-space. The weg-space turned out to play a central role in De Wilde closed graph theorems.

In this paper we show that neither the web nor the web-space is required for this type of closed graph theorem. Its validity depends precisely on the countability and completeness properties of the web-space and not on the intrinsic structure of the web itself.

46A30Open mapping and closed graph theorems; completeness
46A19Generalizations of topological linear spaces