By Marc De Wilde

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**Sample text**

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 -

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).