Foundations of Mathematical Optimization pp 390-411 | Cite as

# Polynomials. Necessary and sufficient conditions of optimality of higher order

## 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 *i* ≠ *i* _{0}, *F*(x _{1} ^{0} , ...,x _{0−I} ^{0} , x_{i0}, 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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