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Semi-norms

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

Abstract

The semi-norm of a vector in a linear space gives a kind of length for the vector. To introduce a topology in a linear space of infinite dimension suitable for applications to classical and modern analysis, it is sometimes necessary to make use of a system of an infinite number of semi-norms. It is one of the merits of the Bourbaki group that they stressed the importance, in functional analysis, of locally convex spaces which are defined through a system of semi-norms satisfying the axiom of separation. If the system reduces to a single semi-norm, the corresponding linear space is called a normed linear space. If, furthermore, the space is complete with respect to the topology defined by this semi-norm, it is called a Banach space. The notion of complete normed linear spaces was introduced around 1922 by S. Banach and N. Wiener independently of each other. A modification of the norm, the quasi-norm in the present book, was introduced by M. Fréchet. A particular kind of limit, the inductive limit,of locally convex spaces is suitable for discussing the generalized functions or the distributions introduced by L. Schwartz, as a systematic development of S. L. Sobolev’s generalization of the notion of functions.

Keywords

Compact Subset Linear Space Triangle Inequality Cauchy Sequence Convex Space 
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.

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References for Chapter I

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Copyright information

© Springer-Verlag Berlin Heidelberg 1968

Authors and Affiliations

  • Kôsaku Yosida
    • 1
  1. 1.Department of MathematicsUniversity of TokyoTokyoJapan

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