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Approximative compactness of linear combinations of characteristic functions. (English) Zbl 1445.41020

We recall that in a normed linear space \(X\), \(B \subset X\) is said to be approximatively compact, if every \(x \notin B\), every minimizing sequence in \(B\) to \(d(x,B)\) has a convergent susbequence with limit in \(B\) (such an element in \(B\) is then a best approximation to \(x\)). For \(1 \leq p \leq \infty\) and for a compact set \(G\) of characteristic functions in \(L^p\), the authors show that the set of \(n\)-fold linear combinations from \(G\) is an approximatively compact set in the set of characteristic fumctions in \(L^p\).

MSC:

41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
41A45 Approximation by arbitrary linear expressions
52A37 Other problems of combinatorial convexity
68T05 Learning and adaptive systems in artificial intelligence
92B20 Neural networks for/in biological studies, artificial life and related topics
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