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General Theory

Chapter
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Part of the Classics in Mathematics book series (CLASSICS, volume 60)

Abstract

Let A be a linear associative algebra over the complex numbers. The algebra A is called a normed algebra if there is associated to each element x a real number ∥x∥, called the norm of x, with the properties:
  1. I

    x∥ ≥ 0 and ∥x∥ = 0 if and only if x = 0;

     
  2. II

    x + y∥ ≤ ∥x∥ + ∥y∥;

     
  3. III

    λx∥ = |λ| ∥x∥, λ a complex number;

     
  4. IV

    xy∥ ≤ ∥x∥ ∥y∥.

     

Keywords

Banach Space Positive Element Polar Decomposition Partial Isometry Strong Operator Topology 
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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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  1. 1.Department of MathematicsNihon UniversitySetagaya-Ku, Tokyo 156Japan

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