The Hahn-Banach Theorems

  • Kôsaku Yosida
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 123)


In a Hilbert space, we can introduce the notion of orthogonal coordinates through an orthogonal base, and these coordinates are the values of bounded linear functionals defined by the vectors of the base. This suggests that we consider continuous linear functionals, in a linear topological space, as generalized coordinates of the space. To ensure the existence of non-trivial continuous linear functionals in a general locally convex linear topological space, we must rely upon the Hahn-Banach extension theorems.


Normed Linear Space Linear Topological Space Continuous Linear Functional Real Linear Space Complex Linear Space 
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References for Chapter IV

  1. [1]
    Banach, S. Théorie des Opérations Linéaires, Warszawa 1932.Google Scholar
  2. [2]
    Bourbaki, N. Espaces Vectoriels Topologiques. Act. Sci. et Ind., nos. 1189, 1229, Hermann 1953–55.Google Scholar
  3. [1]
    Köthe, G. Topologische lineare Räume, Vol. I, Springer 1960.zbMATHGoogle Scholar
  4. [2]
    Mazur, G. Über konvexe Mengen in linearen normierten Räumen. Stud. Math. 5, 70–84 (1933).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1971

Authors and Affiliations

  • Kôsaku Yosida
    • 1
  1. 1.Research Institute for Mathematical SciencesUniversity of KyotoKyotoJapan

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