RT Journal Article
T1 Locally Lipschitz functions, cofinal completeness, and UC spaces
A1 Beer, Gerald
A1 Garrido, M. Isabel
AB Let X, d be a metric space. We find necessary and sufficient conditions on the space for the locally Lipschitz functions to coincide with each of two more restrictive classes of locally Lipschitz functions studied by several authors: the uniformly locally Lipschitz functions and the Lipschitz in the small functions. In the first case, we get the cofinally complete spaces and in the second, the UC spaces. We address this question: to which family of subsets of X, d is the restriction of each function in each class actually Lipschitz? Finally, we determine exactly when the class of uniformly locally Lipschitz functions is uniformly dense in the Cauchy continuous real-valued functions, a class that naturally contains them. In fact, our theorem is valid when the target space is any Banach space. Our density theorem complements the uniform approximation results of Garrido and Jaramillo [12, 13].
PB Elsevier Science
YR 2015
FD 2015-08-04
LK https://hdl.handle.net/20.500.14352/22987
UL https://hdl.handle.net/20.500.14352/22987
LA eng
NO M. Atsuji, Uniform continuity of continuous functions of metric spaces, Pacific J. Math. 8 (1958), 11-16.G. Beer, More about metric spaces on which continuous functions are uniformly continuous, Bull. Austral. Math. Soc. 33 (1986), 397-406.G. Beer, Topologies on closed and closed convex sets, Kluwer Academic Publishers, Dordrecht, Holland, 1993.G. Beer, On metric boundedness structures, Set-valued Anal. 7 (1999), 195-208.G. Beer, Between compactness and completeness, Top. Appl. 155 (2008), 503-514.G. Beer and G. Di Maio, The bornology of cofinally complete subsets, Acta Math. Hungar. 134 (2012), 322-343.G. Beer and M. I. Garrido, Bornologies and locally Lipschitz functions, Bull. Austral. Math. Soc. 90 (2014), 257-263.N. Bourbaki, Elements of mathematics, general topology, Part 1, Hermann, Paris, 1966.G. Di Maio, E. Meccariello, and S. Naimpally, Decompostions of UC spaces, Questions and Answers in Gen. Top. 22 (2004), 13-22.J. Fried and Z. Frolik, A characterization of uniform paracompactness, Proc. Amer. Math. Soc. 89 (1983), 537-540.Z. Frolik, Existence of ∞ partitions of unity, Rend. Sem. Mat. Univ. Politech. Torino 42 (1984), 9-14.M. I. Garrido and J. Jaramillo, Homomorphisms on function lattices, Monatsh. Math. 141 (2004), 127-146.M. I. Garrido and J. Jaramillo, Lipschitz-type functions on metric spaces, J. Math. Anal. Appl. 340 (2008), 282-290.M. I. Garrido and A.S. Mero˜no, New types of completeness in metric spaces, Ann. Acad. Sci. Fennicae 39 (2014), 733-758.J. Hejcman, Boundedness in uniform spaces and topological groups, Czech. Math. J. 9 (1959), 544-563.H. Hogbe-Nlend, Bornologies and functional analysis, North-Holland, Amsterdam, 1977.A. Hohti, On uniform paracompactness, Ann. Acad. Sci. Fenn. Series A Math. Diss. 36 (1981), 1-46.N. Howes, Modern analysis and topology, Springer-Verlag, New York, 1995.T. Jain and S. Kundu, Atsuji completions: equivalent characterizations, Top. Appl. 154 (2007), 28-38.E. Lowen-Colebunders, Function classes of Cauchy continuous functions, Marcel Dekker, New York, 1989.J. Luukkainen, Rings of functions in Lipschitz topology, ann. Acad. Sci. Fenn. Series A. I. Math. 4 (1978-70), 119-135.G. Marino, When is any continuous function Lipschitzian? Extracta Math. 13 (1998), 107-110. A. Monteiro and M. Peixoto, Le nombre de Lebesgue et la continuité uniforme, Portugaliae Math. 10 (1951), 105-113.S. Nadler and T. West, A note on Lebesgue spaces, Topology Proc. 6 (1981), 363-369.W. Pfeffer, The divergence theorem and sets of finite perimeter, CRC Press, Boca Raton, FL, 2012.M. Rice, A note on uniform paracompactness, Proc. Amer. Math. Soc. 62 (1977), 359-362.C. Scanlon, Rings of functions with certain Lipschitz properties, Pacific J. Math. 32 (1970), 197-201.R. Snipes, Functions that preserve Cauchy sequences, Nieuw Archief Voor Wiskunde 25 (1977), 409-422.G. Toader, On a problem of Nagata, Mathematica (Cluj) 20 (1978), 78-79.T. Vroegrijk, Uniformizable and realcompact bornological universes, Appl. Gen. Top. 10 (2009), 277-287.S. Willard, General topology, Addison-Wesley, Reading, MA, 1970.
NO MINECO
DS Docta Complutense
RD 29 abr 2024