Closed Graph Theorems and Webbed Spaces by Marc De Wilde

By Marc De Wilde

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We have sketched above an approach to a characterization of the classes of range spaces. e. when deciding, in concrete examples, wether a space belongs to them or not. Thus it may be more important to exhibit nice subclasses of ~&' large enough to contain those spaces that occur in practice and having reasonable permanence properties. We already mentionned that the set & of all ultrabornological - spaces would be an acceptable class of domain spaces. There do exist interesting subclasses of the corresponding ~g• The first one was introduced by Raikov [87].

2. A TOPOLOGICAL INTERPRETATION The topological meaning of the "completing" condition has been discovered by M. Nakamura [76]. It also appears, for more particular webs, in [89], [90] and [81]. Although not essential for further developments, it should be welcomed by experts, reluctant to use the above naive definition. It moreover clearly exhibits the link between our results and Banach's results 52 [6]. about complete metric groups Since a web is completing if and only if each of its strands is completing, we just look for the topological meaning of a completing sequence.

There is a partial converse to prop. 10, due to Iyahen [60]. However, we quote it here under the additional assumption that F is complete which seems necessary in its proof. 1~ [60). s. 1L L belongs ~ 8F' ~ E belongs to &F. We may assume that L is of codimension one. Let T be a linear map of E into F such that G(T) is closed in Ex F. Its restriction TIL to L has a closed graph in L x F, so it is continuous. Assume that L is dense in E. If x 0 f E, the filter of neighborhoods of x 0 in E induces a Cauchy filter ~ in L.

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