Real Analysis Exchange

Approximations by Lipschitz functions generated by extensions.

Radu Miculescu

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Abstract

We show that, for each pair of metric spaces that has the Lipschitz extension property, every bounded uniformly continuous function can be approximated by Lipschitz functions. The same statement is valid for functions between a locally convex space and $\mathbb{R}^{n}$. In addition, we show that for a locally bounded, convex function $F:X\rightarrow\mathbb{R}^{n}$, where $X$ is a separable Fréchet space, the set of points on which the differential of this function exists is dense in $X$.

Article information

Source
Real Anal. Exchange, Volume 28, Number 1 (2002), 33-41.

Dates
First available in Project Euclid: 12 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.rae/1150118740

Mathematical Reviews number (MathSciNet)
MR1973966

Zentralblatt MATH identifier
1074.41013

Subjects
Primary: 26A16: Lipschitz (Hölder) classes 41A99: None of the above, but in this section

Keywords
Lipschitz functions approximation extension

Citation

Miculescu, Radu. Approximations by Lipschitz functions generated by extensions. Real Anal. Exchange 28 (2002), no. 1, 33--41. https://projecteuclid.org/euclid.rae/1150118740


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References

  • N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces, Studia Math., 57 (1976), 147–190.
  • A. Bressan and A. Cortesi, Lipschitz extensions of convex-valued maps, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 80 (1986), no. 7–12, 530-532. MR 90a:49013.
  • G. Georganopoulos, Sur l'approximation des fonctions continues par des fonctions lipschitziennes, C. R. Acad. Sci. Paris Sér. A, 264 (1967), 319–321. MR 35:5839.
  • J. Luukkainen and J. Väisälä, {Elements of
  • P. Mankiewicz, On the differentiability of Lipschitz mappings in Fréchet spaces, Studia Math., 45 (1973), 15–29. MR 48:9390.
  • E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc, 40 (1934), 837–842.
  • D. Preiss, Differentiability of Lipschitz functions on Banach spaces, J. Funct. Anal., 91 (1990), no. 2, 312–345. MR 91g:46051.
  • H. Rademacher, Über partielle and totale Differenzierbarkeit von Funktionen mehrerer Variabeln and über die Transformation der Doppelintegrale, Math. Ann., 79 (1919), 340–359.
  • S. O. Schönbeck, On the extension of Lipschitz maps, Ark. Mat., 7 (1967), 201–209. MR 35:3297.
  • F. A. Valentine, A Lipschitz condition preserving extension for a vector function, Amer. J. Math., 67 (1945), 83–93.