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Polynomials. Necessary and sufficient conditions of optimality of higher order

  • Diethard Pallaschke
  • Stefan Rolewicz
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 388)

Abstract

In sufficient conditions and necessary conditions of optimality of higher order an essential role is played by polynomials. Let (X 1,‖ · ‖1), (X 2,‖ · ‖2),..., (X n ,‖ · ‖ n ), (Y,‖ · ‖ Y ) be Banach spaces. If it does not lead to any misunderstanding we shall write the norms ‖ · ‖1,..., ‖ · ‖n, ‖ · ‖Y as ‖ · ‖. An n-linear operator is an operator F(x 1,..., x n ) mapping the Cartesian product X 1 ×...; × X n into Y such that for the all variables fixed except one, x i = x i 0 for ii 0, F(x 1 0 , ...,x 0−I 0 , xi0, x 0+I 0 , ..., x n 0 is a linear operator mapping \({X_{{i_0}}}\) into the space Y. In the sequel we shall consider only continuous n-linear operators. When Y is a field of scalars, n-linear operators are called n-linear forms. 2-linear forms will be called also bilinear forms.

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Copyright information

© Springer Science+Business Media Dordrecht 1997

Authors and Affiliations

  • Diethard Pallaschke
    • 1
  • Stefan Rolewicz
    • 2
  1. 1.Institute for Statistics and Mathematical EconomicsUniversity of KarlsruheKarlsruheGermany
  2. 2.Institute of Mathematics of the Polish Academy of SciencesWarsawPoland

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