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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 }$.

References [Enhancements On Off] (What's this?)

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Keywords: Fréchet derivative, Gateaux derivative
Article copyright: © Copyright 1979 American Mathematical Society

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