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.

**[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.

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