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

A characterization of smooth functions defined on a Banach space


Author: Richard M. Hain
Journal: Proc. Amer. Math. Soc. 77 (1979), 63-67
MSC: Primary 58C20; Secondary 26E10
MathSciNet review: 539632
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Abstract: A sufficient condition for a function defined on a Banach space to be $ {C^k}$ is given. This enables us to characterize the $ {C^\infty }$ functions from one Banach space into another Banach space as those functions that, for each positive integer m, have the property that the composition of the function with each $ {C^\infty }$ function from $ {{\mathbf{R}}^m}$ into the domain of the function is $ {C^\infty }$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0539632-8
PII: S 0002-9939(1979)0539632-8
Keywords: Fréchet derivative, Gateaux derivative
Article copyright: © Copyright 1979 American Mathematical Society