## Convergence theorems for best approximations in a nonreflexive Banach space.(English)Zbl 0924.41021

Let $$X$$ be a Banach space, $$A\subset X$$, $$x\in X$$ and $$P_A(x):=\{y_0\in A: d(x,A)=\inf\{\| x-y\|: y\in A\} = \| x-y_0\|\}$$. Denote by $${\mathcal C}$$ the family of all proximinal subsets of $$X$$ (a set is called proximinal if $$P_A(x)\neq\emptyset$$, for every $$x\in X$$).
The author studies continuity properties of the application: (1) $$(x,A)\to P_A(x)$$, $$x\in X$$, $$A\in {\mathcal A}$$, with respect to Wijsman strong convergence in a nonreflexive Banach space.
If $$X$$ is a reflexive Banach space then Mosco convergence implies Wijsman strong convergence (Proposition 2.1) so that the obtained results are extensions to the nonreflexive case of known results in reflexive Banach spaces with respect to Mosco convergence (e.g. M. Tsukada, J. Approximation Theory 40, 301-309 (1984; Zbl 0545.41042)]; N. S. Papageorgiou and D. A. Kandilakis, J. Approximation Theory 49, 41-54 (1987; Zbl 0619.41033)].

### MSC:

 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 41A50 Best approximation, Chebyshev systems

### Keywords:

best approximation

### Citations:

Zbl 0545.41042; Zbl 0619.41033
Full Text:

### References:

 [1] Borwein, J.M.; Fitzpatrick, S., Mosco convergence and the kadec property, Proc. amer. math. soc., 106, 843-851, (1989) · Zbl 0672.46007 [2] Brosowski, B.; Deutsch, F.; Nürnberger, G., Parametric approximation, J. approx. theory, 29, 261-271, (1980) · Zbl 0483.41033 [3] Diestel, J., Geometry of Banach spaces-selected topics, Lecture notes in mathematics, 485, (1975), Springer-Verlag Berlin/Heidelberg/New York [4] Kutzarova, D.; Lin, Bor-Luh, Locallyk, Math. balkanica, 8, 203-210, (1994) · Zbl 0896.46004 [5] Qiyuan, Na, On fully convex and locally fully convex Banach space, Acta math. sci., 10, 327-343, (1990) · Zbl 0734.46009 [6] Chao-Xun, Nan; Jian-Hua, Wang, On the lk-UR and L-kr spaces, Math. proc. Cambridge philos. soc., 104, 521-526, (1988) · Zbl 0673.46008 [7] Panda, B.B.; Kapoor, O.P., A generalization of local uniform convexity of the norm, J. math. anal. appl., 52, 300-308, (1975) · Zbl 0314.46014 [8] Papageorgion, N.S.; Kandilakis, D.A., Convergence in approximation and nonsmooth analysis, J. approx. theory, 49, 41-54, (1987) · Zbl 0619.41033 [9] Rainwater, J., Local uniform convexity of Day’s norm onc0γ, Proc. amer. math. soc., 22, 335-339, (1969) · Zbl 0185.37602 [10] Singer, I., Best approximation in normed-linear spaces by elements of linear subspaces, (1973), Springer-Verlag Berlin [11] Sullivan, F., A generalization of uniformly rotund Banach spaces, Canad. J. math., 31, 628-646, (1979) · Zbl 0422.46011 [12] Tsukada, M., Convergence of best approximations in a smooth Banach space, J. approx. theory, 40, 301-309, (1984) · Zbl 0545.41042 [13] Jianhua, Wang, Some results on the continuity of metric projections, Math. appl., 8, 80-84, (1995) · Zbl 0949.46501 [14] Jianhua, Wang; Chao-Xun, Nan, On the convergence of ε-approximation, Banach space theory and its application, (1996), Wuhan Univ. Press [15] Jianhua, Wang; Musan, Wang, Compactly locally fully convex spaces, Kexue tongbao, 36, 796, (1991) [16] Xintai, Yu, On LKUR spaces, Chinese ann. math. (ser. B), 6, 465-469, (1985) · Zbl 0595.46022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.