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More about Euclidean Vector Spaces

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Part of the Problem Books in Mathematics book series (PBM)

Abstract

A Euclidean vector space E is a real, finite-dimensional vector space (whose dimension will always be denoted by n), endowed with a positive definite symmetric bilinear form ϕ (i.e. ϕ(x, x) > 0 for all x ≠ 0). We write ϕ(x, y) as (x|y), and we call this number the scalar product (or inner product) of x and y. The number
$$\left\| x \right\| = \sqrt {\varphi \left( {x,x} \right)} $$
is called the norm of x. Two vectors x and y are called orthogonal if (x|y) = 0.

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Copyright information

© Marcel Berger 1984

Authors and Affiliations

  1. 1.U.E.R. de Mathematique et InformatiqueUniversité Paris VIIParis, Cedex 05France
  2. 2.Centre National de la Recherche ScientifiqueParisFrance
  3. 3.P.U.K.GrenobleFrance

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