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Extension of polynomials defined on subspaces

Published online by Cambridge University Press:  16 March 2010

MAITE FERNÁNDEZ-UNZUETA  and
ÁNGELES PRIETO
MAITE FERNÁNDEZ-UNZUETA
Affiliation:
Centro de Investigación en Matemáticas (CIMAT), A.P. 402, Guanajuato, 36000, Gto., Mexico. e-mail: maite@cimat.mx
ÁNGELES PRIETO
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain. e-mail: angelin@mat.ucm.es
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Abstract

Let k ∈ ℕ and let E be a Banach space such that every k-homogeneous polynomial defined on a subspace of E has an extension to E. We prove that every norm one k-homogeneous polynomial, defined on a subspace, has an extension with a uniformly bounded norm. The analogous result for holomorphic functions of bounded type is obtained. We also prove that given an arbitrary subspace F ⊂ E, there exists a continuous morphism φk, F: (kF) → (kE) satisfying φk, F(P)|F = P, if and only E is isomorphic to a Hilbert space.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 148 , Issue 3 , May 2010 , pp. 505 - 518
Copyright
Copyright © Cambridge Philosophical Society 2010

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