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Scalar Products and Orthogonality

  • Serge Lang
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

Let V be a vector space over a field K. A scalar product on V is an association which to any pair of elements v, w of V associates a scalar, denoted by <v, w>, or also v·w, satisfying the following properties:
  1. SP 1.

    We have <v, w> = <w, v> for all v, wV.

     
  2. SP 2.
    If, u, v, w are elements of V, then
    $$<u,v+ w>=<u,v>+<u,w>$$
    .
     
  3. SP 3.
    If xK, then
    $$<xu,v>= x<u,v>$$
    and
    $$u,xv>=x<u,v>$$
     

Keywords

Vector Space Scalar Product Orthonormal Basis Dual Space Orthogonal Basis 
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 Science+Business Media New York 1987

Authors and Affiliations

  • Serge Lang
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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