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