Nonlinear approximation in uniformly smooth Banach spaces

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 F : R n → X F:{R^n} \to X be a homeomorphism onto M = F ( R n ) M = F({R^n}) . Suppose ∇ F ( a ) \nabla F(a) exists for each a in X, ∇ F \nabla F is continuous as a function of a, and ∇ F ( a ) ⋅ R n \nabla F(a) \cdot {R^n} 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 Y = R n Y = {R^n} .] The paper is concluded with a few remarks on Chebyshev sets.