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J. Math. Anal. Appl. 326 (2007) 13701378 www.elsevier.com/locate/jmaa
 

Summary: J. Math. Anal. Appl. 326 (2007) 1370­1378
www.elsevier.com/locate/jmaa
Smooth approximation of Lipschitz functions
on Riemannian manifolds
D. Azagra
, J. Ferrera, F. López-Mesas, Y. Rangel
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense, 28040 Madrid, Spain
Received 10 February 2006
Available online 2 May 2006
Submitted by R.M. Aron
Abstract
We show that for every Lipschitz function f defined on a separable Riemannian manifold M (possibly
of infinite dimension), for every continuous :M (0,+), and for every positive number r > 0, there
exists a C smooth Lipschitz function g :M R such that |f (p) - g(p)| (p) for every p M and
Lip(g) Lip(f ) + r. Consequently, every separable Riemannian manifold is uniformly bumpable. We also
present some applications of this result, such as a general version for separable Riemannian manifolds of
Deville­Godefroy­Zizler's smooth variational principle.
© 2006 Elsevier Inc. All rights reserved.
Keywords: Lipschitz function; Riemannian manifold; Smooth approximation
1. Introduction and main results

  

Source: Azagra Rueda, Daniel - Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid

 

Collections: Mathematics