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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Quasi-differentiable functions of Banach spaces

Author: Victor Goodman
Journal: Proc. Amer. Math. Soc. 30 (1971), 367-370
MSC: Primary 46.45; Secondary 57.00
MathSciNet review: 0284811
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Abstract: Nonzero Fréchet differentiable functions with bounded support do not exist on certain real separable Banach spaces. As a result, the class of differentiable functions on such spaces is too small to be useful. For example, the class of differentiable functions on certain spaces does not separate disjoint closed subsets of the space. It is shown that this separation problem does not arise if Fréchet differentiability is replaced by the weaker condition of quasi-differentiability. Furthermore, it is shown that any bounded uniformly continuous function on a real separable Banach space is the uniform limit of quasi-differentiable functions.

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Additional Information

PII: S 0002-9939(1971)0284811-3
Keywords: Differentiable functions on Banach spaces, calculus on Banach spaces, smoothing operators
Article copyright: © Copyright 1971 American Mathematical Society

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