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On infinite-dimensional topological groups

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On infinite-dimensional topological groups Séminaire d’analyse fonctionnelle École Polytechnique P. ENFLO On infinite-dimensional topological groups Séminaire d’analyse fonctionnelle (Polytechnique) (1977-1978), exp. no 10 et 11, p. 1-11. <> © 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 ( 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 SEMINAIRE S U R L A G E 0 M E T R I E DES ESPACES D E B A N A C If 1977-1978 ON INFINITE-DIMENSIONAL TOPOLOGICAL GROUPS P. ENFLO (Institut Mittag-Leffler) PCOLE POLYTECHNIQUE CENTRE DE MATHISMATIQUES PLATEAU DE PALAISEAU - 91128 PALAISEAU CEDEX Téléphone : 941.82.00 . Paste N· T6]ex : ECOLEX 691596 P Expos6s No X-XI 1 i-?0 Janvier 1978 X-XI.1 I - INTRODUCTION In this seminar we will discuss infinite-dimensional topologi- cal groups. We will do this mainly in the spirit of Hilbert’s fifth problem-In 1900 Hilbert asked among other questions the following : Is every topological group that is locally homeomorphic to R , a Lie n group ? (For definitions we refer to [7j). Around 1950 this question was given an affirmative answer as the result of the joint efforts of several researchers (see [6J for a presentation of the solution). Before this result was proved it had been proved that even weak dif- ferentiability assumptions on the group operations imply that a group is a Lie group. In 1938 G. Birkhoff r 1-1 proved that if in a locally Euclidean group (x,y)-xy is continuously differentiable t.hen the group is a Lie

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