# Closed Graph Theorems and Webbed Spaces by Marc De Wilde

By Marc De Wilde

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Extra resources for Closed Graph Theorems and Webbed Spaces

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L. 1. l. ,x 1 )B o C 2CB. Bi C 29 m 1 (*). Take now 1 9 " = c o ( 9 u 29 ' ) ; n it is a bornivorous disk of E, such that 9" L = e. It is obvious that a bornivorous disk of E induces a bornivorous disk of L, thus we have proved that Lb has the topology induced by E 0 • This implies as above that L is bornological if E is bornological. See [110] for the ultrabornological case. Let us now turn to criteria for dual spaces to be bornological. As far as applications are concerned, the main result is the following, due to L.

Let E- E be 41 such that (x 0 ,x'> = 1 and denote by T the projection x - x - x 0 of E onto L. Assume first that x' is locally bounded. Then T maps bounded sets of E into bounded sets of L, If 9 is a bornivorous disk of L, T- 1 9 is a bornivorous disk of E and 9 = L n T- 1 e. If 9 1 is a bornivorous disk of E, 9 1 n L is bornivorous in L. Hence Lb has If E is bornological, the topology induced by Eb. L is thus bor- nological too. Assume now that x' bounded sequence x ~ X m is not locally bounded.

S. equipped with the finest locally convex to- pology. Hence, for instance, prop. 16 is still true when U is the class of barrelled spaces, ultrabornological spaces, etc. PROPOSITION Ilo3o17o [Mahowald, 7 ] . j ! 'iFis the class of all Banach spaces, led spaces. ~U ~ &u are equal to the class of all barrel- Every barrelled space belongs to ~U , hence to &U • Indeed, let E be barrelled and R be a linear relation of E into a Banach space F such that ~(R) E X = E and G(R) is closed in F. For every x E- E, if xRy, then xRy 1 ~ y 1 E- y + R(O) • Moreover, R(O) is a vector subspace of F and it is closed, since 22 {0} R(O) = G(R) n ( {0} x R) X • Thus R induces a linear map T of E into F/R(O), where F/R(O) is another Banach space, and G(T) has a closed graph, since G(T) {0} with = I ( {0} G(R) x R(O) R(O)) X ' C G(R).