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Complexifications of real Banach spaces, polynomials and multilinear maps. (English) Zbl 0945.46010
In the first part of the paper, various procedures to obtain a norm in the complexification of a real Banach space are compared. General schemes of such procedures are introduced. In the second part, the norm of the corresponding complexification of polynomials (homogeneous and nonhomogeneous) is estimated. In particular, estimates of Chebyshev type are obtained where the norm of the (first or second) leading coefficient of the complex polynomial is estimated in terms of the norm of the real polynomial. Similar estimates for multilinear maps (not necessarily symmetric) are obtained. The involved constants are shown to be best possible.

MSC:
46B99 Normed linear spaces and Banach spaces; Banach lattices
46G25 (Spaces of) multilinear mappings, polynomials
47H60 Multilinear and polynomial operators
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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