Rendiconti del Circolo Matematico di Palermo

, Volume 37, Issue 3, pp 416–430 | Cite as

Polynomials in topological vector spaces over valued fields

  • Dinamérico P. Pombo


We consider the vector space of continuousm-homogeneous polynomials between topological vector spaces over a non-trivially valued field of characteristic zero and certain natural vector topologies on such spaces, and we prove polynomial versions of certain well known theorems of the linear theory of locally convex spaces.


Topological Vector Space Closed Unit Ball Null Sequence Polynomial Version Balance Neighborhood 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Aron R., Schottenloher M.,Compact Holomorphic Mappings on Banach Spaces and the Approximation Property, J. Func. Analysis21 (1976), 7–30.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Bochnak J., Siciak J.,Polynomials and multilinear mappings in topological vector spaces, Studia Math.39 (1971), 59–76.zbMATHMathSciNetGoogle Scholar
  3. [3]
    Bourbaki N.,Topologie générale, chapitre 10, Hermann (1967).Google Scholar
  4. [4]
    Bourbaki N.Espaces vectoriels topologiques, chapitres 1 et 2, Hermann (1966).Google Scholar
  5. [5]
    Bourbaki N.Espaces vectoriels topologiques, chapitres 3, 4 et 5, Hermann (1967).Google Scholar
  6. [6]
    Dineen S.,Complex Analysis in Locally Convex Spaces, North-Holland Math. Studies57 (1981).Google Scholar
  7. [7]
    Ingleton A. W.,The Hahn-Banach theorem for non-Archimedean valued fields, Proc. Cambridge Phil. Society48 (1952), 41–45.zbMATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    Kakutani S.,A proof of Schauder's Theorem, J. Math. Soc. Japan3 (1951), 228–231.zbMATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    Mazur S., Orlicz W.,Grundlegende Eigenschaften der polynomischen Operationen, Studia Math.5 (1934), 50–68, 179–189.Google Scholar
  10. [10]
    Monna A. F.,Espaces localement convexes sur un corps valué, Proc. Kon. Ned. Akad. v. Wetensch A62 (1959), 391–405.MathSciNetzbMATHGoogle Scholar
  11. [11]
    Monna A.F.,Analyse non-archimédienne, Ergebnisse der Math.56 (1970), Springer-Verlag.Google Scholar
  12. [12]
    Nachbin L.,Topology on spaces of holomorphic mappings, Ergebnisse der Math.47 (1969), Springer-Verlag.Google Scholar
  13. [13]
    Pombo D.P., Jr.,On polynomial classification of locally convex spaces, Studia Math.78 (1984), 39–57.zbMATHMathSciNetGoogle Scholar
  14. [14]
    Pombo D.P., Jr.,A non-archimedean analogue of the Alaoglu-Bourbaki theorem, Bull. Inst. Math. Ac. Sinica12 (1984), 205–210.zbMATHMathSciNetGoogle Scholar
  15. [15]
    Pombo D.P., Jr.,The Alaoglu-Bourbaki theorem for continuous polynomials, An. Acad. Bras. Ciências57 (1985), 153–154.zbMATHMathSciNetGoogle Scholar
  16. [16]
    Prolla J.B.,Approximation of Vector Valued Functions, North-Holland Math. Studies25 (1977).Google Scholar
  17. [17]
    Prolla J.B.,Topics in Functional Analysis over Valued Division Rings, North-Holland Math. Studies77 (1982).Google Scholar
  18. [18]
    Ramis J.P.,Sous-ensembles analytiques d'une variété banachique complexe, Ergebnisse der Math.53 (1970), Springer-Verlag.Google Scholar
  19. [19]
    Van Rooij A.C.M.,Non-archimedean functional analysis, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., (1978).Google Scholar
  20. [20]
    Schauder J.,Über lineare vollstetige Funktionaloperationen, Studia Math.2 (1930), 183–196.Google Scholar

Copyright information

© Springer 1988

Authors and Affiliations

  • Dinamérico P. Pombo
    • 1
  1. 1.Instituto de MatemáticaUniversidade Federal do Rio de JaneiroRio de JaneiroBrasil

Personalised recommendations