# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 ${\cal C}$ the family of all proximinal subsets of $X$ (a set is called proximinal if $P_A(x)\ne\emptyset$, for every $x\in X$). The author studies continuity properties of the application: (1) $(x,A)\to P_A(x)$, $x\in X$, $A\in {\cal A}$, 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. {\it M. Tsukada}, J. Approximation Theory 40, 301-309 (1984; Zbl 0545.41042)]; {\it N. S. Papageorgiou} and {\it D. A. Kandilakis}, J. Approximation Theory 49, 41-54 (1987; Zbl 0619.41033)].

##### MSC:
 41A65 Abstract approximation theory 41A50 Best approximation, Chebyshev systems
##### Keywords:
best approximation
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) · Zbl 0307.46009 [4] Kutzarova, D.; Lin, Bor-Luh: Locallyk. Math. balkanica 8, 203-210 (1994) [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 $onc0{\Gamma}$. 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) · Zbl 0271.41026 [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 ${\epsilon}$-approximation. (1996) [15] Jianhua, Wang; Musan, Wang: Compactly locally fully convex spaces. Kexue tongbao 36, 796 (1991) · Zbl 0833.46006 [16] Xintai, Yu: On LKUR spaces. Chinese ann. Math. (Ser. B) 6, 465-469 (1985) · Zbl 0595.46022