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On some non-archimedean normed linear spaces. II

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On some non-archimedean normed linear spaces. II COMPOSITIO MATHEMATICA PIERREROBERT On some non-archimedean normed linear spaces. II Compositio Mathematica, tome 19, no 1 (1968), p. 16-27. <http://www.numdam.org/item?id=CM_1968__19_1_16_0> © Foundation Compositio Mathematica, 1968, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). 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 http://www.numdam.org/ 16 On some non-Archimedean normed linear spaces II by Pierre Robert Introduction This paper is the second of a series published under the same title and numbered I, II, .... The reader is assumed to be familiar with the definitions, notations and results of Part I. This Part II is devoted to the study of V-spaces (see Part I, Introduction). In the last Section, V-algebras are introduced and results to be used in the theory of operators on V-spaces (Part IV) are stated. 1. Definitions A systematic study of non-Archimedean normed linear spaces has been made by A. F. Monna ([24], [25]). Other references are [3], [12], [17]. Monna obtains interesting results when the norm range of the non-Archimedean normed linear space is assumed to have at most one accumulation point: 0. We shall retain this assumption. In most of his work, Monna requires that the valuation of the field of scalars be non-trivial; this, of course, is impossible in the case of a valued space. DEFINITION 1.1. A V-space X is a pseudo-valued or a valued space which is complete in its norm topology and for which there exists a set of integers c(X)) and a real number p &#x3E; 1, such that (II.1) 03A9(X

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