## Convergence of continuous linear functionals and their level sets.(English)Zbl 0662.46015

Let $$X$$ be a real Banach space with continuous dual $$X^*$$. We characterize both norm and weak* convergence of sequences in $$X^*$$ to a nonzero limit in terms of the convergence of the level sets of the linear functionals. When $$X$$ is reflexive, norm convergence is equivalent to the Mosco convergence of level sets. Using this fact, we show that Mosco convergence of sequences of closed convex sets in a reflexive space may be properly stronger than pointwise convergence of the distance functions for the sets in the sequence.
Reviewer: G.Beer

### MSC:

 46B10 Duality and reflexivity in normed linear and Banach spaces 54B20 Hyperspaces in general topology 46B20 Geometry and structure of normed linear spaces
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### References:

 [1] H.Attouch, Variational convergence for functions and operators. Boston 1984. · Zbl 0561.49012 [2] J. P.Aubin, Applied abstract analysis. New York 1977. · Zbl 0393.54001 [3] M.Baronti and P.Papini, Convergence of sequences of sets. In: Methods of functional analysis in approximation theory, ISNM 76, Basel 1986. · Zbl 0606.54006 [4] G. Beer, Metric spaces with nice closed balls and distance functions for closed sets. Bull. Austral. Math. Soc.35, 81-96 (1987). · Zbl 0588.54014 [5] G. Beer, On Mosco convergence of convex sets. Bull. Austral. Math. Soc.38, 239-253 (1988). · Zbl 0669.52002 [6] A. L. Brown, A rotund reflexive space having a subspace of codimension two with a discontinuous metric projection. Michigan Math. J.21, 145-151 (1974). · Zbl 0275.46016 [7] J. Fell, A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space. Proc. Amer. Math. Soc.13, 472-476 (1962). · Zbl 0106.15801 [8] S. Francaviglia, A. Lechicki andS. Levi, Quasi-uniformization of hyperspaces and convergence of nets of semicontinuous multifunctions. J. Math. Anal. Appl.112, 347-370 (1985). · Zbl 0587.54003 [9] Z. Frolik, Concerning topological convergence of sets. Czech. Math. J.10, 168-180 (1960). · Zbl 0095.37103 [10] R. Holmes, Approximating best approximations. Nieuw Arch. Wisk.14, 106-113 (1966). [11] J. Joly, Une famille de topologies sur l’ensemble des fonctions convexes pour lesquelles la polarit? est bicontinue. J. Math. Pures Appl.52, 421-441 (1973). · Zbl 0282.46005 [12] E.Klein and A.Thompson, Theory of correspondences. Toronto 1984. · Zbl 0556.28012 [13] K.Kuratowski, Topology, Vol. 1. New York 1966. [14] S. Levi andA. Lechicki, Wijsman convergence in the hyperspace of a metric space. Boll. Un. Mat. Ital. (7)1-B, 439-451 (1987). · Zbl 0655.54007 [15] U. Mosco, Convergence of convex sets and of solutions of variational inequalities. Adv. in Math.3, 510-585 (1969). · Zbl 0192.49101 [16] U. Mosco, On the continuity of the Young-Fenchel transform. J. Math. Anal. Appl.35, 518-535 (1971). · Zbl 0253.46086 [17] A.Robertson and W.Robertson, Topological vector spaces. Cambridge 1973. · Zbl 0251.46002 [18] R. T.Rockafellar and R.Wets, Variational systems, an introduction. In: Multifunctions and integrands, G. Salinetti, ed., LNM1091, Berlin-Heidelberg-New York 1984. [19] G. Salinetti andR. Wets, Convergence of convex sets in finite dimensions. SIAM Rev.21, 18-33 (1979). · Zbl 0421.52003 [20] Y. Sonntag, Convergence au sens de Mosco; th?orie et applications ? l’approximation des solutions d’in?quations. Th?se d’?tat. Universit? de Provence, Marseille 1982. [21] M. Tsukada, Convergence of best approximations in a smooth Banach space. J. Approx. Theory40, 301-309 (1984). · Zbl 0545.41042 [22] D. Walkup andR. Wets, Convergence of some convex-cone valued mappings. Proc. Amer. Math. Soc.18, 229-235 (1967). · Zbl 0145.38004
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