Über die Existenz gemeinsamer Elemente bester Approximation bezüglich zweier Normen (Q1071202)

This paper deals with the possibility of existence of best approximation elements, simultaneously with respect to two norms for all the elements of a class of subspaces. The main result is: Let E be a vector space endowed with two norms \(\| \cdot \|_ i\), \(i=1,2\). The following propositions are equivalent: (i) For every \(\| \cdot \|_ 1\)- closed n-codimensional (n\(\in {\mathbb{N}})\) subspace G of E and for every \(x\in E\) one has (*) \(P^ 1_ G(x)\subset P^ 2_ G(x)\). (ii) There exists \(k\in N\) such that for every k-dimensional subspace G of E and for every \(x\in E\) (*) holds. (iii) For every 1-dimensional subspace G of E and for every \(x\in E\) one has \(P^ 1_ G(x)\cap P^ 2_ G(x)\neq \emptyset.\) (iv) There exists \(\alpha >0\) such that \(\| x\|_ 1=\alpha \| x\|_ 2\) for all \(x\in E\).