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Higher order Gateaux smooth bump functions on Banach spaces

Published online by Cambridge University Press:  17 April 2009

David P. McLaughlin  and
Jon D. Vanderwerff
David P. McLaughlin
Affiliation:
Department of MathematicsKwantlen CollegeSurrey BCCanada V3T 5H8
Jon D. Vanderwerff
Affiliation:
Department of Mathematics and StatisticsSimon Fraser UniversityBurnaby BCCanada V5A 1S6
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Abstract

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For Г uncountable and p ≥ 1 odd, it is shown ℓp(г) admits no continuous p-times Gateaux differentiable bump function. A space is shown to admit a norm with Hölder derivative on its sphere if it admits a bounded bump function with uniformly directionally Hölder derivative. Some results on smooth approximation are obtained for spaces that admit bounded uniformly Gateaux differentiable bump functions.

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 51 , Issue 2 , April 1995 , pp. 291 - 300
Copyright
Copyright © Australian Mathematical Society 1995

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