##
**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 FrĂ©chet 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.

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 FrĂ©chet 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.

### MSC:

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.) |