On the uniform approximation of Cauchy continuous functions
| dc.contributor.author | Beer, Gerald | |
| dc.contributor.author | Garrido, M. Isabel | |
| dc.date.accessioned | 2023-06-18T06:54:06Z | |
| dc.date.available | 2023-06-18T06:54:06Z | |
| dc.date.issued | 2016 | |
| dc.description.abstract | In the context of real-valued functions defined on metric spaces, it is known that the locally Lipschitz functions are uniformly dense in the continuous functions and that the Lipschitz in the small functions - the locally Lipschitz functions where both the local Lipschitz constant and the size of the neighborhood can be chosen independent of the point - are uniformly dense in the uniformly continuous functions. Between these two basic classes of continuous functions lies the class of Cauchy continuous functions, i.e., the functions that map Cauchy sequences in the domain to Cauchy sequences in the target space. Here, we exhibit an intermediate class of Cauchy continuous locally Lipschitz functions that is uniformly dense in the real-valued Cauchy continuous functions. In fact, our result is valid when our target space is an arbitrary Banach space. | |
| dc.description.department | Depto. de Álgebra, Geometría y Topología | |
| dc.description.faculty | Fac. de Ciencias Matemáticas | |
| dc.description.refereed | TRUE | |
| dc.description.sponsorship | Ministerio de Economía y Competitividad (MINECO) | |
| dc.description.status | pub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/38178 | |
| dc.identifier.doi | 10.1016/j.topol.2016.04.017 | |
| dc.identifier.issn | 0166-8641 | |
| dc.identifier.officialurl | http://www.sciencedirect.com/science/article/pii/S0166864116300578 | |
| dc.identifier.relatedurl | http://www.sciencedirect.com/ | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/24538 | |
| dc.issue.number | 1 | |
| dc.journal.title | Topology and its Applications | |
| dc.language.iso | eng | |
| dc.page.final | 9 | |
| dc.page.initial | 1 | |
| dc.publisher | Elsevier Science | |
| dc.relation.projectID | MTM2012-34341 | |
| dc.rights.accessRights | restricted access | |
| dc.subject.cdu | 517.53 | |
| dc.subject.keyword | Cauchy continuous function | |
| dc.subject.keyword | Cauchy-Lipschitz function | |
| dc.subject.keyword | Lipschitz in the small function | |
| dc.subject.keyword | Locally Lipschitz function | |
| dc.subject.keyword | Primary | |
| dc.subject.keyword | Secondary | |
| dc.subject.keyword | Uniform approximation | |
| dc.subject.ucm | Funciones (Matemáticas) | |
| dc.subject.unesco | 1202 Análisis y Análisis Funcional | |
| dc.title | On the uniform approximation of Cauchy continuous functions | |
| dc.type | journal article | |
| dc.volume.number | 208 | |
| dcterms.references | [1] M. Atsuji, Uniform continuity of continuous functions of metric spaces, Pac. J. Math. 8 (1958) 11–16. [2] G. Beer, More about metric spaces on which continuous functions are uniformly continuous, Bull. Aust. Math. Soc. 33 (1986) 397–406. [3] G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer Academic Publishers, Dordrecht, Holland, 1993. [4] G. Beer, Between compactness and completeness, Topol. Appl. 155 (2008) 503–514. [5] G. Beer, G. Di Maio, Cofinal completeness of the Hausdorff metric topology, Fundam. Math. 208 (2010) 75–85. [6] G. Beer, M.I. Garrido, Bornologies and locally Lipschitz functions, Bull. Aust. Math. Soc. 90 (2014) 257–263. [7] G. Beer, M.I. Garrido, Locally Lipschitz functions, cofinal completeness, and UC spaces, J. Math. Anal. Appl. 428 (2015) 804–816. [8] J. Borsik, Mappings preserving Cauchy sequences, Čas. Pěst. Mat. 113 (1988) 280–285. [9] G. Di Maio, E. Meccariello, S. Naimpally, Decompositions of UC spaces, Quest. Answ. Gen. Topol. 22 (2004) 13–22. [10] Z. Frolik, Existence of ∞ partitions of unity, Rend. Semin. Mat. Univ. Politech. Torino 42 (1984) 9–14. [11] M.I. Garrido, J. Jaramillo, Homomorphisms on function lattices, Monatshefte Math. 141 (2004) 127–146. [12] M.I. Garrido, J. Jaramillo, Lipschitz-type functions on metric spaces, J. Math. Anal. Appl. 340 (2008) 282–290. [13] J. Heinonen, Lectures on Analysis in Metric Spaces, Springer, New York, 2001. [14] A. Hohti, On uniform paracompactness, Ann. Acad. Sci. Fenn. Ser. A, Math. Diss. 36 (1981) 1–46. [15] N. Howes, Modern Analysis and Topology, Springer-Verlag, New York, 1995. [16] T. Jain, S. Kundu, Atsuji spaces: equivalent conditions, Topol. Proc. 30 (2006) 301–325. [17] T. Jain, S. Kundu, Atsuji completions: equivalent characterizations, Topol. Appl. 154 (2007) 28–38. [18] E. Lowen-Colebunders, Function Classes of Cauchy Continuous Functions, Marcel Dekker, New York, 1989. [19] J. Luukkainen, Rings of functions in Lipschitz topology, Ann. Acad. Sci. Fenn., Ser. A 1 Math. 4 (1978–1979) 119–135. [20] J. Luukkainen, J. Väisälä, Elements of Lipschitz topology, Ann. Acad. Sci. Fenn., Ser. A 1 Math. 3 (1977) 85–122. [21] R. Miculescu, Approximation of continuous functions by Lipschitz functions, Real Anal. Exch. 26 (2000–2001) 449–452. [22] S. Nadler, T. West, A note on Lesbesgue spaces, Topol. Proc. 6 (1981) 363–369. [23] C. Scanlon, Rings of functions with certain Lipschitz properties, Pac. J. Math. 32 (1970) 197–201. [24] R. Snipes, Functions that preserve Cauchy sequences, Nieuw Arch. Wiskd. 25 (1977) 409–422. [25] S. Willard, General Topology, Addison-Wesley, Reading, MA, 1970. | |
| dspace.entity.type | Publication |
Download
Original bundle
1 - 1 of 1