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The Quarterly Journal of Mathematics Advance Access originally published online on March 30, 2007
The Quarterly Journal of Mathematics 2007 58(2):221-228; doi:10.1093/qmath/ham010
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© 2007. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

SMOOTH NORMS AND APPROXIMATION IN BANACH SPACES OF THE TYPE C(K)

Petr Hájek1 {dagger} and Richard Haydon2,{ddagger}

1 Mathematical Institute, Czech Academy of Sciences, Zitná 25, Praha 11567, Czech Republic
2 Brasenose College, Oxford OX1 4AJ, UK

{ddagger} Corresponding author. E-mail: richard.haydon{at}brasenose.oxford.ac.uk

Received 3 November 2006;
  Abstract

Two results are proved about the Banach space X = C(K), where K is compact and Hausdorff. The first concerns smooth approximation: let m be a positive integer or {infty}; we show that if there exists on X a non-zero function of class Cm with bounded support, then all continuous real-valued functions on X can be uniformly approximated by functions of class Cm. The second result is that if X admits a norm, equivalent to the supremum norm, with locally uniformly convex dual norm, then X also admits an equivalent norm that is of class C{infty} (except at 0).

{dagger} E-mail: hajek{at}math.cas.cz


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