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 converse of Taylor's theorem for functions on Banach spaces


Author: S. Dayal
Journal: Proc. Amer. Math. Soc. 65 (1977), 265-273
MSC: Primary 58C20; Secondary 46G05
MathSciNet review: 0448394
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The main results are the local representation theorems associating the local weak n-Taylor series expansion of a function defined on a Banach space to a local n-Taylor series expansion of the coefficients. These theorems are used to prove a converse of Taylor's theorem which uses weaker hyptohesis than used by others. Another useful application of the above results is done in [2] to study a class of functions called n-convex functions.


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

  • [1] Ralph Abraham and Joel Robbin, Transversal mappings and flows, An appendix by Al Kelley, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0240836
  • [2] S. Dayal, Local representation and higher differentiability of n-convex operators on Banach spaces (to appear).
  • [3] -, Local representation of functions on normed linear spaces, Ph.D Thesis, Case Western Reserve Univ., Cleveland, Ohio, 1972.
  • [4] -, A converse of Taylor's theorem for functions on locally convex and topological linear spaces (to appear).
  • [5] J. Dieudonné, Foundations of modern analysis, Pure and Applied Mathematics, Vol. X, Academic Press, New York-London, 1960. MR 0120319
  • [6] Georges Glaeser, Étude de quelques algèbres tayloriennes, J. Analyse Math. 6 (1958), 1–124; erratum, insert to 6 (1958), no. 2 (French). MR 0101294
  • [7] F. B. Hildebrand, Introduction to numerical analysis, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1956. MR 0075670
  • [8] J. Kopeć and J. Musielak, On the estimation of the norm of the 𝑛-linear symmetric operation, Studia Math. 15 (1955), 29–30. MR 0073944
  • [9] E. B. Leach, A note on inverse function theorems, Proc. Amer. Math. Soc. 12 (1961), 694–697. MR 0126146, 10.1090/S0002-9939-1961-0126146-9
  • [10] J. Marcinkiewicz and A. Zygmund, On the differentiability of functions and summability of trignometric series, Fund. Math. 26 (1936), 1-43.
  • [11] Robert M. McLeod, Mean value theorems for vector valued functions, Proc. Edinburgh Math. Soc. (2) 14 (1964/1965), 197–209. MR 0185052
  • [12] M. Z. Nashed, Differentiability and related properties of nonlinear operators: Some aspects of the role of differentials in nonlinear functional analysis, Nonlinear Functional Anal. and Appl. (Proc. Advanced Sem., Math. Res. Center, Univ. of Wisconsin, Madison, Wis., 1970) Academic Press, New York, 1971, pp. 103–309. MR 0276840
  • [13] A. E. Taylor, Addition to the theory of polynomials in normed linear spaces, Tôhoku Math. J. 44 (1938), 302-318.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58C20, 46G05

Retrieve articles in all journals with MSC: 58C20, 46G05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1977-0448394-2
Keywords: Fréchet differential, strong differential, multilinear functions, discrete differences, convex property, convex set, local representations, weak and strong n-Taylor series expansion
Article copyright: © Copyright 1977 American Mathematical Society