Theorem 13. Let E be a Baire space, F the countable reduced inductive limit of strictly webbed spaces (F„) and f:E-^F a linear mapping with closed graph. Then there is an index p such that f(E) is contained in Fp and f: E -^ Fp is tinuous.

As a corollary one gets that a strictly webbed Baire space is a Fréchet space.

A stronger localization theorem for webbed spaces has been obtained recently by Valdivia [83]: If/: £ -> F is a linear mapping with closed graph, E being a Baire space and F webbed, there is a subspace H of F and a topology s on H finer that the induced topology of F such that/i£) is contained in Я, (Я, s) is a Fréchet space and /: E -> (Я, s) is continuous.

As corollaries one gets: (i) Let F be a webbed space. If F is the locally convex hull of Baire spaces, then F is ultrabornological. (ii) Let F be a webbed space. If F is the locally convex hull of a sequence of Baire spaces, then F is an (LF)-space.

The webbed spaces enjoy good hereditary properties. Valdivia has shown [87]: "'E ®aFya = e, 7i) is webbed iff one of the two following conditions is satisfied: (i) E is webbed and dim F ^ Kq, (ii) F is webbed and dim E S ^o"-

Tt was an open problem whether De Wilde's closed graph theorem contained Schwartz's closed graph theorem. Since the projective tensor product of Suslin spaces is a Suslin space [87], it is enough to consider Suslin spaces E and F of non countable infinite dimension to conclude that there are Suslin spaces which are not webbed and thus De Wilde's theorem does not contain Schwarz's result [87].

In the category of Banach spaces the closed graph theorem is equivalent to the open-mapping theorem and this is not necessarily true in the category of locally convex spaces. In order to give a closed graph theorem extending Banach's classical result beyond the scope of metrizable spaces, Ptak introduced Б^-complete spaces as spaces F such that every quasi-continuous linear mapping f: E -> F, E being any space, with closed graph is continuous and therefore one has: "Let E be a barreled space, F a Б^-complete space, then any linear mapping/: F -> F with closed graph is continuous."

The property of being J5^-complete is connected with the Krein-Smul'jan property and with completeness by means of the following characterization: "A space E is ß^-complete iff every dense subspace of its topological dual E' which intersects every closed absolutely convex F-equicontinuous set in weakly closed sets is closed in (F', cr(F', F)). According to the Krein-Smul'jan theorem every Fréchet space is Б^-complete (thus, the range class for Ptak's closed graph theorem contains all Fréchet spaces) and every Б^-complete space is complete due to Ptak-ColUns's characterization of completeness. In order to give an open-mapping-theorem, Ptak introduced J?-complete spaces which are characterized by means of "A space E is Б-complete iff every subspace of its topological dual E' which intersects every closed absolutely convex F-equicontinuous set in weakly closed sets is closed in (F', cr(F', F))." If a space is Б-complete, every separated quotient is Б-complete and every closed subspace is also Б-complete. Moreover, clearly every Б-complete

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