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Higher order Gateaux smooth bump functions on Banach spaces

1995; Cambridge University Press; Volume: 51; Issue: 2 Linguagem: Inglês

10.1017/s000497270001412x

1755-1633

David McLaughlin, Jon D. Vanderwerff,

Optimization and Variational Analysis

For Г uncountable and p ≥ 1 odd, it is shown ℓ p (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.