Mosco convergence and the Kadec property

Authors:
Jonathan M. Borwein and Simon Fitzpatrick

Journal:
Proc. Amer. Math. Soc. **106** (1989), 843-851

MSC:
Primary 46B20; Secondary 52A05, 54C60

MathSciNet review:
969313

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the relationship between Wijsman convergence and Mosco convergence for sequences of convex sets. Our central result is that Mosco convergence and Wijsman convergence coincide for sequences of convex sets if and only if the underlying space is reflexive with the dual norm having the Kadec property.

**[At]**H. Attouch,*Variational convergence for functions and operators*, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR**773850****[Au]**Jean-Pierre Aubin,*Applied abstract analysis*, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. Exercises by Bernard Cornet and Hervé Moulin; Translated from the French by Carole Labrousse; Pure and Applied Mathematics. MR**470034****[Be]**G. Beer,*Convergence of continuous linear functionals and their level sets*, Arch. Math. (in press).**[Bo-Fi]**J. M. Borwein and S. P. Fitzpatrick,*Existence of nearest points in Banach spaces*, Canadian Journal of Mathematics (in press).**[Da]**Mahlon M. Day,*Normed linear spaces*, 3rd ed., Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21. MR**0344849****[GGS]**J. R. Giles, D. A. Gregory, and Brailey Sims,*Geometrical implications of upper semi-continuity of the duality mapping on a Banach space*, Pacific J. Math.**79**(1978), no. 1, 99–109. MR**526669****[Ho]**Richard B. Holmes,*A course on optimization and best approximation*, Lecture Notes in Mathematics, Vol. 257, Springer-Verlag, Berlin-New York, 1972. MR**0420367****[Ko]**S. V. Konjagin,*Approximation properties of closed sets in Banach spaces and the characterization of strongly convex spaces*, Dokl. Akad. Nauk SSSR**251**(1980), no. 2, 276–280 (Russian). MR**565493****[La]**Ka Sing Lau,*Almost Chebyshev subsets in reflexive Banach spaces*, Indiana Univ. Math. J.**27**(1978), no. 5, 791–795. MR**0510772****[Mo1]**Umberto Mosco,*Convergence of convex sets and of solutions of variational inequalities*, Advances in Math.**3**(1969), 510–585. MR**0298508****[Mo2]**Umberto Mosco,*On the continuity of the Young-Fenchel transform*, J. Math. Anal. Appl.**35**(1971), 518–535. MR**0283586****[Sa-We]**Gabriella Salinetti and Roger J.-B. Wets,*On the convergence of sequences of convex sets in finite dimensions*, SIAM Rev.**21**(1979), no. 1, 18–33. MR**516381**, 10.1137/1021002**[So]**Y. Sonntag,*Convergence au sens de Mosco; théorie et applications à l'approximation des solutions d'inéquations*, Thèse d'État. Université de Provence, Marseille, 1982.**[Ts]**Makoto Tsukada,*Convergence of best approximations in a smooth Banach space*, J. Approx. Theory**40**(1984), no. 4, 301–309. MR**740641**, 10.1016/0021-9045(84)90003-0**[Wi]**R. A. Wijsman,*Convergence of sequences of convex sets, cones and functions. II*, Trans. Amer. Math. Soc.**123**(1966), 32–45. MR**0196599**, 10.1090/S0002-9947-1966-0196599-8

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
46B20,
52A05,
54C60

Retrieve articles in all journals with MSC: 46B20, 52A05, 54C60

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1989-0969313-4

Article copyright:
© Copyright 1989
American Mathematical Society