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Best approximation in spaces of bounded linear operators. (English) Zbl 0826.41025

The problem of best approximation in a normed linear space \(X\) by elements of a nonempty set \(V\) is formulated as follows. Given \(x\) in \(X\), find an element \(v_0\) in \(V\) for which the lower bound \[ \text{dist} (x,V) = \inf \bigl\{ |x - v |: v \in V \bigr\} \] is attained. The element \(v_0\) is called an element of best approximation. In this paper the author considers the following problems (1) existence of element of best approximation, (2) uniqueness of element of best approximation, (3) characterization of elements of best approximation, (4) estimation of the constant \(\text{dist} (x,V)\), (5) construction of algorithms for obtaining elements of best approximation in the case where \(X = L(B,D)\), the space of continuous linear mappings from a normed vector space \(B\) into a normed vector space \(D\). Sections II.1, II.2 and Chapter 3 contain new material.
For earlier work on these problems, see W. Odyniec and G. Lewicki, Minimal projections in Banach spaces, Lect. Notes in Math. No. 1449 (Springer-Verlag, 1990).

MSC:

41A50 Best approximation, Chebyshev systems
41A35 Approximation by operators (in particular, by integral operators)
41A52 Uniqueness of best approximation
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
46B99 Normed linear spaces and Banach spaces; Banach lattices
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
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