Garrido Carballo, M. Isabel and Jaramillo Aguado, Jesús Ángel (2008) Lipschitz-type functions on metric spaces. Journal of Mathematical Analysis and Applications, 340 (1). pp. 282-290. ISSN 0022-247X
![[img]](/style/images/fileicons/application_pdf.png) | PDF Restricted to Repository staff only until 2020. 152Kb |
Official URL: http://www.sciencedirect.com/science/article/pii/S0022247X0701044X
Abstract
In order to find metric spaces X for which the algebra Lip*(X) of bounded Lipschitz functions on X determines the Lipschitz structure of X, we introduce the class of small-determined spaces. We show that this class includes precompact and quasi-convex metric spaces. We obtain several metric characterizations of this property, as well as some other characterizations given in terms of the uniform approximation and the extension of uniformly continuous functions. In particular we show that X is small-determined if and only if every uniformly continuous real function on X can be uniformly approximated by Lipschitz functions.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | Banach-Stone theorem; Lipschitz functions; small-determined metric space; uniform approximation |
| Subjects: | Sciences > Mathematics > Functional analysis and Operator theory |
| ID Code: | 16329 |
| References: | R.F. Arens, J. Eells, On embedding uniform and topological spaces, Pacific J. Math. 6 (1956) 397–403. Y. Benyamini, J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Amer. Math. Soc. Colloq. Publ., vol. 48, Amer. Math. Soc., Providence, RI, 2000. D. Burago, Y. Burago, S. Ivanov, A Course in Metric Geometry, Grad. Stud. Math., vol. 33, Amer. Math. Soc., Providence, RI, 2001. J. Bustamante, J.R. Arrazola, Homomorphisms on Lipschitz spaces, Monatsh. Math. 129 (2000) 25–30. V.A. Efremovich, The geometry of proximity I, Sb. Math. 31 (1952) 189–200. M.I. Garrido, J.A. Jaramillo, A Banach–Stone theorem for uniformly continuous functions, Monatsh. Math. 131 (2000) 189–192. M.I. Garrido, J.A. Jaramillo, Variations on the Banach–Stone theorem, Extracta Math. 17 (2002) 351–383. M.I. Garrido, J.A. Jaramillo, Homomorphisms on function lattices, Monatsh. Math. 141 (2004) 127–146. M.I. Garrido, F. Montalvo, Countable covers and uniform closure, Rend. Istit. Mat. Univ. Trieste 30 (1999) 91–102. L. Géher, Über fortsetzungs und approximationprobleme für stetige abbildungen von mestrichen raumen, Acta Sci. Math. (Szeged) 20 (1959) 48–66. L. Gillman, J. Jerison, Rings of Continuous Functions, Springer-Verlag, New York, 1976. G. Godefroy, N.J. Kalton, Lipschitz-free Banach spaces, Studia Math. 159 (2003) 121–141. M. Katetov, On real-valued functions on topological spaces, Fund. Math. 38 (1951) 85–91. R. Levy, M.D. Rice, Techniques and examples in U-embedding, Topology Appl. 22 (1986) 157–174. R. Levy, M.D. Rice, U-embedded subsets of normed linear spaces, Proc. Amer. Math. Soc. 97 (1986) 727–733. J. Luukainen, Rings of functions in Lipschitz topology, Ann. Acad. Sci. Fen. 4 (1978/1979) 119–135. E.J. McShane, Extensions of range of functions, Bull. Amer. Math. Soc. 40 (1934) 837–842. D.R. Sherbert, Banach algebras of Lipschitz functions, Pacific J. Math. 13 (1963) 1387–1399. Li Pi Su, Algebraic properties if certain rings of continuous functions, Pacific J. Math. 27 (1968) 175–191. N. Weaver, Lattices of Lipschitz functions, Pacific J. Math. 164 (1994) 179–193. N. Weaver, Lipschitz Algebras, World Scientific, Singapore, 1999. |
| Deposited On: | 12 Sep 2012 13:06 |
| Last Modified: | 12 Sep 2012 13:06 |
Repository Staff Only: item control page
