Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1979-0539632-8
MathSciNet review: 539632
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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?)

  • [1] R. Abraham and J. Robbin, Transversal mappings and flows, Benjamin, New York, 1967. MR 0240836 (39:2181)
  • [2] F. Albrecht and H. Diamond, A converse of Taylor's theorem, Indiana Univ. Math. J. 21 (1971/72), 347-350. MR 0320744 (47:9278)
  • [3] K. T. Chen, Global calculus, mimeographed class notes, 1976.
  • [4] -, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977), 831-879. MR 0454968 (56:13210)
  • [5] S. Lang, Differentiable manifolds, Addison-Wesley, Reading, Mass., 1972.
  • [6] J. T. Schwartz, Nonlinear functional analysis, Gordon and Breach, New York, 1969. MR 0433481 (55:6457)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58C20, 26E10

Retrieve articles in all journals with MSC: 58C20, 26E10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1979-0539632-8
Keywords: Fréchet derivative, Gateaux derivative
Article copyright: © Copyright 1979 American Mathematical Society

American Mathematical Society