Strong unicity criterion in some space of operators. (English) Zbl 0785.41023

Lewicki, Grzegorz

Commentat. Math. Univ. Carol. 34, No. 1, 81-87 (1993).

Summary: Let \(X\) be a finite-dimensional Banach space and let \(Y\subset X\) be a hyperplane. Let \({\mathcal L}_ Y= \{L\in {\mathcal L}(X,Y)\): \(L|_ Y=0\}\). In this note, we present sufficient and necessary conditions on \(L_ 0\in {\mathcal L}_ Y\) being a strongly unique best approximation for given \(L\in {\mathcal L}(X)\). Next, we apply this characterization to the case of \(X=\ell_ \infty^ n\) and to generalization of Theorem I.1.3 from Wl. Odyniec and G. Lewicki [Minimal projections in Banach spaces, Lect. Notes Math. 1449 (1990)].

MSC:

41A65	Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
41A52	Uniqueness of best approximation
41A35	Approximation by operators (in particular, by integral operators)
41A50	Best approximation, Chebyshev systems

Keywords:

strongly unique best approximation