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The polarization constant of finite dimensional complex spaces is one

Part of: Basic linear algebra Measures, integration, derivative, holomorphy General theory of linear operators Nonlinear functional analysis

Published online by Cambridge University Press:  04 March 2021

VERÓNICA DIMANT ,
DANIEL GALICER  and
JORGE TOMÁS RODRÍGUEZ
VERÓNICA DIMANT
Affiliation:
Departamento de Matemática y Ciencias, Universidad de San Andrés, Vito Dumas 284, (B 1644BID) Victoria, Buenos Aires, Argentina and CONICET. e-mail : vero@udesa.edu.ar
DANIEL GALICER
Affiliation:
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Intendente Güiraldes 2160, (1428) Buenos Aires, Argentina and CONICET. e-mail : dgalicer@dm.uba.ar
JORGE TOMÁS RODRÍGUEZ
Affiliation:
Departamento de Matemática and NUCOMPA, Facultad de Cs. Exactas, Universidad Nacional del Centro de la Provincia de Buenos Aires, Pinto 399, (7000) Tandil, Argentina and CONICET. e-mail : jtrodrig@dm.uba.ar
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Abstract

The polarization constant of a Banach space X is defined as

\[{\text{c}}(X){\text{ }}{\text{ }}\mathop {\lim }\limits_{k \to \infty } {\text{ }}\sup {\text{c}}{(k,X)^{\frac{1}{k}}},\]

where \[{\text{c}}(k,X)\] stands for the best constant \[C > 0\] such that \[\mathop P\limits^ \vee \leqslant CP\] for every k-homogeneous polynomial \[P \in \mathcal{P}{(^k}X)\]. We show that if X is a finite dimensional complex space then \[{\text{c}}(X) = 1\]. We derive some consequences of this fact regarding the convergence of analytic functions on such spaces.

The result is no longer true in the real setting. Here we relate this constant with the so-called Bochnak’s complexification procedure.

We also study some other properties connected with polarization. Namely, we provide necessary conditions related to the geometry of X for \[c(2,X) = 1\] to hold. Additionally we link polarization constants with certain estimates of the nuclear norm of the product of polynomials.

MSC classification

Primary: 46G25: (Spaces of) multilinear mappings, polynomials 47A07: Forms (bilinear, sesquilinear, multilinear)
Secondary: 15A69: Multilinear algebra, tensor products 46T25: Holomorphic maps
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 172 , Issue 1 , January 2022 , pp. 105 - 123
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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