Linear Topological Spaces over Non-Archimedean Valued Fields

  • A. F. Monna
Conference paper


I propose to give a survey of the results of the study of linear topological spaces over non-archimedean valued fields. Proofs of theorems will not be given.


Orthogonal Basis Extension Property Normed Linear Space Linear Topological Space Compact Topological Space 
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Notes and References

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    This notion was introduced by A. W. Ingleton in his paper The Hahn-Banach theorem for non archimedean valued fields, Proc. Cambridge Phil. Soc. 48, 41–45 (1952). See the remarks on the binary intersection property in the paper by Nachbin [7].MathSciNetMATHCrossRefGoogle Scholar
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    See the references given in Van Tiel [14].Google Scholar
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    I mention also a characterization of reflexivity in terms of problems of shortest distance; see Bor-Lux Lin, Distance sets in normed vector spaces, Nieuw Archief voor Wiskunde 14, 23–30 (1966).MATHGoogle Scholar
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    In mention here the work of Grauert und Remmert, Nichtarchimedische Funktionentheorie, Festschr. Gedächtnisfeier K. Weierstrass, 393–476, Köln: Westdeutscher Verlag 1966.Google Scholar
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    For projections see the older literature. For linear operators in normed spaces see : Monna, A. F.: Lineaire functionaalvergelijkingen in niet-archimedische Banach-ruimtes. Ned. Akad. v. Wetensch. A 52, 654–661 (1943).MathSciNetMATHGoogle Scholar
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    See also : Serre, J.-P.: Endomorphisms complètement continus des espaces de Banach p-adiques. Inst. H. E. Sci, no. 12 (1962);Google Scholar
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    See also A. C. H. van Rooij: Proc. Kon. Ned. Akad. v. Wetensch. A LXX. 220–228 (1967), where results on the existence of invariant means are obtained. The only property of K relevant to the existence of a means is the characteristic of the residue class field.Google Scholar
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  38. Srinivasan, V. K.: On certain summation processes in the p-adic field. Proc. Kon. Ned. Akad. v. Wetensch. A 68, 319–325 (1965).MATHGoogle Scholar
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    Robert, P.: On some non-archimedean normed linear spaces (to appear).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1967

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  • A. F. Monna

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