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The Quarterly Journal of Mathematics 2005 56(1):13-20; doi:10.1093/qmath/hah020
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© The Author (2005). Published by Oxford University Press. All rights reserved. For Permissions please email: journals.permissions@oupjournals.org

C1-fine approximation of functions on Banach spaces with unconditional basis

Daniel Azagra1 *, Javier Gómez Gil1 §, Jesús A. Jaramillo1 ¶, Mauricio Lovo1 and Robb Fry2 {ddagger}

1 Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense, 28040 Madrid, Spain, 2 Department of Mathematics and Computer Science, St Francis Xavier University, PO Box 5000, Antigonish, Nova Scotia B2G 2W5, Canada

We show that if X is a Banach space having an unconditional basis and a Cp-smooth Lipschitz bump function, then for every C1-smooth function f from X into a Banach space Y, and for every continuous function {varepsilon} : X -> (0, {infty}), there exists a Cp-smooth function g : X -> Y such that ||f – g|| ≤ {varepsilon} and ||f' – g'|| ≤ {varepsilon}.

Received 16 September 2003.

* E-mail: daniel_azagra{at}mat.ucm.es

{ddagger} E-mail: rfry{at}stfx.ca

§ E-mail: Javier_Gomez{at}mat.ucm.es

E-mail: jaramil{at}mat.ucm.es


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