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 ⁰ _i for i ≠ i ₀, F(x ⁰₁ , ...,x ⁰_0−I , x_i0, x ⁰_0+I , ..., x ⁰_n 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.