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Weakly compact sets in $L^1/H^1_0$

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Abstract

Weakly compact sets in L1/H10 Séminaire d’analyse fonctionnelle École Polytechnique F. DELBAEN Weakly compact sets in L1/H10 Séminaire d’analyse fonctionnelle (Polytechnique) (1977-1978), exp. no 8, p. 1-4. <http://www.numdam.org/item?id=SAF_1977-1978____A7_0> © Séminaire d’analyse fonctionnelle (École Polytechnique), 1977-1978, tous droits réservés. L’accès aux archives du séminaire d’analyse fonctionnelle implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation com- merciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ SEMINAIRE SUR L A G E 0 M E T R I E D E S E S P A C E S D E B A N A C 11 1977-1978 WEAKLY COMPACT SETS IN L1/H1o F. DELBAEN (Université de Bruxelles) ECOLE POLYTECHNIQUE CENTRE DE MATHÉMATIQUES PLATEAU DE PALAISEAU . 91128 PALAISEAU CEDEX Téléphone : 941.82.00 . Poste N8 ECOLEX 691596 F Expose No VIII 16 D6cembre 1977 VIII.1 By A we mean a uniform algebra in ttic sense of T.W. Camelin [ 4~ , i.e. there is a compact Hausdorff space X such tliat A C C (X) , 1 E A and A separates the points of X. If ~ : : A ~ ~ is a nonzero, multiplicative, linear functional then M denotes the set of positive representing measures on X. More precisely M = a positive measure on X and J f for all f 6 A} . We will suppose that M is a weakly compact set in the space of all measures on X. In this case it is easily seen that there is m E M such that all other measures in M are W 4 absolutely continuous with respect to m (f.i. a slight fication of the proof given in Dunford-Schwartz [3] p. 307 already gives this result). By H 00 we mean the Hardy space which is the weak star closure of t% in L"(m) where m is the dominant measure mentioned before. The predual of Hoo is L 1

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