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Nuclear spaces of maximal diametral dimension

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Nuclear spaces of maximal diametral dimension COMPOSITIO MATHEMATICA CHRISTIAN FENSKE EBERHARD SCHOCK Nuclear spaces ofmaximal diametral dimension Compositio Mathematica, tome 26, no 3 (1973), p. 303-308. <> © Foundation Compositio Mathematica, 1973, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// implique l’accord avec les conditions générales d’utilisation ( Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la pré- sente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques 303 NUCLEAR SPACES OF MAXIMAL DIAMETRAL DIMENSION by Christian Fenske and Eberhard Schock COMPOSITIO MATHEMATICA, Vol. 26, Fasc. 3, 1973, pag. 303-308 Noordhoff International Publishing Printed in the Netherlands The diametral dimension 0394(E) of a locally convex vector space E is known to be a measure for the nuclearity of E. Therefore it is of in- terest to characterize the class S2 of those locally convex vector spaces, the diametral dimension of which is maximal. We show that the class S2 has the same stability properties as the class X of all nuclear spaces, and characterize the members of Q, that are contained in the smallest stability class, by a property of their bornology. At first let us define what we mean by a stability class: DEFINITION. (a) A stability class is a class of locally convex vector spaces, which is closed under the operations of forming (Si) completions (S2) subspaces (S3) quotients by closed subspaces (S4) arbitrary products (SS) countable direct sums (S6) tensor products (S7) isomorphic images. (b) If E is a locally convex vector space, we denote by u(E), the stability class of E, the smallest stability class containing

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