Convergence of best approximations in a smooth Banach space. (English) Zbl 0545.41042

Tsukada, Makoto

J. Approximation Theory 40, 301-309 (1984).

Let X be a reflexive and strictly convex Banach space. Then for any non- empty closed convex subsets \(C\subseteq X\) and \(x\in X\) there exists a unique best approximation \(p(x| C)\) of x by elements of C. Whenever the sequence \(\{x_ n\}\subseteq X\) which weakly converges to some \(x\in X\) and \(\| x_ n\| \to \| x\|\) as \(n\to \infty\) necessarily converges to x in norm, X is said to have Property (H). If X further has Property (H), then the mapping \(x\mapsto p(x| C)\) is continuous in norm. In this paper, the continuity of mapping \(C\mapsto p(x| C)\) is investigated with the non-parametralized method. Let \(\{C_ n\}\) be a sequence of non-empty closed convex subsets of X. The following is a main theorem. If X has Property (H), then for any \(x\in X\) the sequence \(\{p(x| C_ n)\}\) converges to \(p(x| \lim C_ n)\) in norm whenever lim \(C_ n\) exists and is not empty. Conversely, if X is finite dimensional or has Fréchet differentiable norm, then lim \(C_ n\) (the strong limit of \(\{C_ n\}\) in Mosco’s sense) exists and is not empty whenever the sequence \(\{p(x| C_ n)\}\) converges in norm for every \(x\in X\). Using the limit of closed convex sets, a measurability condition of closed convex set valued functions can be defined. By the above result, in a certain Banach space, the condition is equivalent to some measurability conditions which are known in the theory of multi- valued functions.

MSC:

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

Keywords:

unique best approximation; non-parametralized method; multi-valued functions

References:

[1]	Brosowski, B.; Deutsch, F.; Nürnberger, G., Parametric approximation, J. Approx. Theory, 29, 261-277 (1980) · Zbl 0483.41033
[2]	Brunk, H. D., Conditional expectation given a σ-lattice and applications, Ann. Math. Statist., 36, 1339-1350 (1965) · Zbl 0144.39503
[3]	Cudia, D. F., Rotundity, (Proc. Symp. Pure Math., 7 (1963)), 73-97 · Zbl 0141.11901
[4]	Day, M. M., Normed Linear Spaces (1973), Springer-Verlag: Springer-Verlag Berlin · Zbl 0268.46013
[5]	Hille, E.; Phillips, R. S., Functional Analysis and Semigroups (1957), Amer. Math. Soc: Amer. Math. Soc Providence, R. I · Zbl 0078.10004
[6]	Himmelberg, C. J., Measurable relations, Fund. Math., 87, 53-72 (1975) · Zbl 0296.28003
[7]	Kudō, H., A note on the strong convergence of σ-algebras, Ann. Prob., 2, 76-83 (1974) · Zbl 0275.60007
[8]	Mosco, U., Convergence of convex sets and of solutions of variational inequalities, Adv. in Math., 3, 510-585 (1969) · Zbl 0192.49101
[9]	Rao, M. M., Prediction sequences in smooth Banach spaces, Ann. Inst. Henri Poincaré, 8, 319-332 (1972) · Zbl 0251.60034
[10]	Singer, I., Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces (1973), Springer-Verlag: Springer-Verlag Berlin
[11]	Tsukada, M., Convergence of closed convex sets and σ-fields, Z. Wahrsch. Verw. Gebiete, 62, 137-146 (1983) · Zbl 0488.60005
[12]	Tsukada, M., The strong limit of von Neumann subalgebras with conditional expectations (1983), preprint