Uniform continuity on bounded sets and the Attouch-Wets topology
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- by Gerald Beer and Anna Di Concilio PDF
- Proc. Amer. Math. Soc. 112 (1991), 235-243 Request permission
Abstract:
Let ${\text {CL(}}X{\text {)}}$ be the nonempty closed subsets of a metrizable space $X$. If $d$ is a compatible metric, the metrizable Attouch-Wets topology ${\tau _{aw}}(d)$ on ${\text {CL(}}X{\text {)}}$ is the topology of uniform convergence of distance functionals associated with elements of ${\text {CL(}}X{\text {)}}$ on bounded subsets of $X$. The main result of this paper shows that two compatible metrics $d$ and $\rho$ determine the same Attouch-Wets topologies if and only if they determine the same bounded sets and the same class of functions that are uniformly continuous on bounded sets.References
- H. Attouch, Variational convergence for functions and operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 773850 H. Attouch, R. Lucchetti, and R. Wets, The topology of the $\rho$-Hausdorff distance, Ann. Mat. Pura Appl. (to appear). H. Attouch and R. Wets, Quantitative stability of variational systems. I. The epigraphical distance, Working paper, IIASA, Laxenburg, Austria, 1988. D. Azé, On some metric aspects of set convergence, preprint. D. Azé and J.-P. Penot, Operations on convergent families of sets and functions, AVAMAC report, Perpignan, 1987.
- Dominique Azé and Jean-Paul Penot, Recent quantitative results about the convergence of convex sets and functions, Functional analysis and approximation (Bagni di Lucca, 1988) Pitagora, Bologna, 1989, pp. 90–110. MR 1001572
- Gerald Beer, Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance, Proc. Amer. Math. Soc. 95 (1985), no. 4, 653–658. MR 810180, DOI 10.1090/S0002-9939-1985-0810180-3
- Gerald Beer, Metric spaces with nice closed balls and distance functions for closed sets, Bull. Austral. Math. Soc. 35 (1987), no. 1, 81–96. MR 875510, DOI 10.1017/S000497270001306X
- Gerald Beer, On Mosco convergence of convex sets, Bull. Austral. Math. Soc. 38 (1988), no. 2, 239–253. MR 969914, DOI 10.1017/S0004972700027519
- Gerald Beer, On the Young-Fenchel transform for convex functions, Proc. Amer. Math. Soc. 104 (1988), no. 4, 1115–1123. MR 937844, DOI 10.1090/S0002-9939-1988-0937844-8
- Gerald Beer, Convergence of continuous linear functionals and their level sets, Arch. Math. (Basel) 52 (1989), no. 5, 482–491. MR 998621, DOI 10.1007/BF01198356
- Gerald Beer, Conjugate convex functions and the epi-distance topology, Proc. Amer. Math. Soc. 108 (1990), no. 1, 117–126. MR 982400, DOI 10.1090/S0002-9939-1990-0982400-8
- Gerald Beer and Jonathan M. Borwein, Mosco convergence and reflexivity, Proc. Amer. Math. Soc. 109 (1990), no. 2, 427–436. MR 1012924, DOI 10.1090/S0002-9939-1990-1012924-9
- Gerald Beer and Roberto Lucchetti, Convex optimization and the epi-distance topology, Trans. Amer. Math. Soc. 327 (1991), no. 2, 795–813. MR 1012526, DOI 10.1090/S0002-9947-1991-1012526-X
- Jonathan M. Borwein and Simon Fitzpatrick, Mosco convergence and the Kadec property, Proc. Amer. Math. Soc. 106 (1989), no. 3, 843–851. MR 969313, DOI 10.1090/S0002-9939-1989-0969313-4
- C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR 0467310 L. Contesse and J.-P. Penot, Continuity of polarity and conjugacyfor the epi-distance topology, preprint.
- Sebastiano Francaviglia, Alojzy Lechicki, and Sandro Levi, Quasiuniformization of hyperspaces and convergence of nets of semicontinuous multifunctions, J. Math. Anal. Appl. 112 (1985), no. 2, 347–370. MR 813603, DOI 10.1016/0022-247X(85)90246-X
- K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
- Ernest Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182. MR 42109, DOI 10.1090/S0002-9947-1951-0042109-4
- Umberto Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Math. 3 (1969), 510–585. MR 298508, DOI 10.1016/0001-8708(69)90009-7
- Umberto Mosco, On the continuity of the Young-Fenchel transform, J. Math. Anal. Appl. 35 (1971), 518–535. MR 283586, DOI 10.1016/0022-247X(71)90200-9
- Somashekhar Amrith Naimpally, Graph topology for function spaces, Trans. Amer. Math. Soc. 123 (1966), 267–272. MR 192466, DOI 10.1090/S0002-9947-1966-0192466-4
- S. A. Naimpally and B. D. Warrack, Proximity spaces, Cambridge Tracts in Mathematics and Mathematical Physics, No. 59, Cambridge University Press, London-New York, 1970. MR 0278261
- Gabriella Salinetti and Roger J.-B. Wets, On the relations between two types of convergence for convex functions, J. Math. Anal. Appl. 60 (1977), no. 1, 211–226. MR 479398, DOI 10.1016/0022-247X(77)90060-9 Y. Sonntag, Convergence au sens de Mosco; théorie et applications à l’approximation des solutions d’inéquations, Thèse d’Etat. Université de Provence, Marseille, 1982.
- Angus Ellis Taylor and David C. Lay, Introduction to functional analysis, 2nd ed., John Wiley & Sons, New York-Chichester-Brisbane, 1980. MR 564653
- Makoto Tsukada, Convergence of best approximations in a smooth Banach space, J. Approx. Theory 40 (1984), no. 4, 301–309. MR 740641, DOI 10.1016/0021-9045(84)90003-0
- David W. Walkup and Roger J.-B. Wets, Continuity of some convex-cone-valued mappings, Proc. Amer. Math. Soc. 18 (1967), 229–235. MR 209806, DOI 10.1090/S0002-9939-1967-0209806-6
- R. A. Wijsman, Convergence of sequences of convex sets, cones and functions. II, Trans. Amer. Math. Soc. 123 (1966), 32–45. MR 196599, DOI 10.1090/S0002-9947-1966-0196599-8
- Stephen Willard, General topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0264581
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 112 (1991), 235-243
- MSC: Primary 54B20
- DOI: https://doi.org/10.1090/S0002-9939-1991-1033956-1
- MathSciNet review: 1033956