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 \((\ell cs)\). 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.


46A30 Open mapping and closed graph theorems; completeness (including \(B\)-, \(B_r\)-completeness)
46A19 Other “topological” linear spaces (convergence spaces, ranked spaces, spaces with a metric taking values in an ordered structure more general than \(\mathbb{R}\), etc.)