Nonlinear approximation in uniformly smooth Banach spaces

Authors:
Edward R. Rozema and Philip W. Smith

Journal:
Trans. Amer. Math. Soc. **188** (1974), 199-211

MSC:
Primary 41A65

DOI:
https://doi.org/10.1090/S0002-9947-1974-0330875-5

MathSciNet review:
0330875

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: John R. Rice [*Approximation of functions*. Vol. II, Addison-Wesley, New York, 1969] investigated best approximation from a nonlinear manifold in a finite dimensional, smooth, and rotund space. The authors define the curvature of a manifold by comparing the manifold with the unit ball of the space and suitably define the ``folding'' of a manifold. Rice's Theorem 11 extends as follows: Theorem. Let *X be a uniformly smooth Banach space, and* *be a homeomorphism onto* . *Suppose* *exists for each a in X*, *is continuous as a function of a, and* *has dimension n. Then, if M has bounded curvature, there exists a neighborhood of M each point of which has a unique best approximation from M*. A variation theorem was found and used which locates a critical point of a differentiable functional defined on a uniformly rotund space *Y*. [See M. S. Berger and M. S. Berger, *Perspectives in nonlinearity*, Benjamin, New York, 1968, p. 58ff. for a similar result when .] The paper is concluded with a few remarks on Chebyshev sets.

**[1]**V. I. Averbuh and O. G. Smoljanov,*Differentiation theory in linear topological spaces*, Uspehi Mat. Nauk**22**(1967), no. 6 (138), 201-260 = Russian Math. Surveys**22**(1967), no. 6, 201-258. MR**36**#6933. MR**0223886 (36:6933)****[2]**M. S. Berger and M. S. Berger,*Perspectives in nonlinearity. An introduction to nonlinear analysis*, Benjamin, New York, 1968. MR**40**#4971. MR**0251744 (40:4971)****[3]**D. F. Cudia,*The geometry of Banach spaces. Smoothness*, Trans. Amer. Math. Soc.**110**(1964), 284-314. MR**29**#446. MR**0163143 (29:446)****[4]**J. Dieudonné,*Foundations of modern analysis*, Pure and Appl. Math., vol. 10, Academic Press, New York, 1960. MR**22**#11074. MR**0120319 (22:11074)****[5]**I. C. Gohberg and A. S. Markus,*Two theorems on the gap between subspaces of a Banach space*, Uspehi Mat. Nauk**14**(1959), no. 5 (89), 135-140. (Russian) MR**22**#5880. MR**0115077 (22:5880)****[6]**J. R. Rice,*Approximation of functions*. Vol. II.*Nonlinear and multivariate theory*, Addison-Wesley, Reading, Mass., 1969. MR**39**#5989. MR**0244675 (39:5989)****[7]**I. Singer,*Best approximation in normed vector spaces by elements of vector subspaces*, Editura Academiei Republicii Socialiste România, Bucharest, 1967; English transl., Die Grundlehren der math. Wissenschaften, Band 171, Springer-Verlag, New York and Berlin, 1970. MR**38**#3677;**42**#4937. MR**0235368 (38:3677)****[8]**L. P. Vlasov,*Čebyšev sets and some generalizations of them*, Mat. Zametki**3**(1968), 59-69. (Russian) MR**37**#3329. MR**0227745 (37:3329)****[9]**D. E. Wulbert,*Continuity of metric projections*, Trans. Amer. Math. Soc.**134**(1968), 335-341. MR**38**#472. MR**0232146 (38:472)****[10]**-,*Uniqueness and differential characterization of approximations from manifolds of functions*, Amer. J. Math.**93**(1971), 350-366. MR**45**#4036. MR**0294968 (45:4036)****[11]**-,*Nonlinear approximation with tangential characterization*, Amer. J. Math.**93**(1971), 718-730. MR**45**#4037. MR**0294969 (45:4037)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
41A65

Retrieve articles in all journals with MSC: 41A65

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0330875-5

Keywords:
Approximation,
nonlinear approximation,
nonlinear functional analysis,
uniformly smooth Banach space

Article copyright:
© Copyright 1974
American Mathematical Society