Quasi-differentiable functions of Banach spaces

Goodman, Victor

doi:10.1090/S0002-9939-1971-0284811-3

Quasi-differentiable functions of Banach spaces
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Proc. Amer. Math. Soc. 30 (1971), 367-370 Request permission

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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