Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Extension of polynomials defined on subspaces

MAITE FERNÁNDEZ-UNZUETAa1 and ÁNGELES PRIETOa2

a1 Centro de Investigación en Matemáticas (CIMAT), A.P. 402, Guanajuato, 36000, Gto., Mexico. e-mail: maite@cimat.mx

a2 Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain. e-mail: angelin@mat.ucm.es

Abstract

Let k xs2208 xs2115 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 xs2282 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.

(Received April 30 2009)

(Revised October 30 2009)

(Online publication March 16 2010)