The unicity of best approximation in a space of compact operators (Q5894044)

Editorial comment: Verbatim the same paper was published by the same author in [Math. Scand. 108, No.~1, 146--160 (2011; Zbl 1239.41010)] as well. Let \(Y\) be a closed subspace of the formed space \(X\). \(Y\) is called Chebyshev in \(X\), if for each \(x\in X\) there exists unique \(y_0\in Y\), such that \(\| x- y_0\|= \text{inf}\{\| x- y\|: y\in Y\}\). A \(k\)- dimensional subspace \(Y\) is called an interpolating subspace if and only if the only element of \(Y\) that can be anhilated by \(k\)-independent extreme points of the unit ball of the dual is the zero element. The main object of the paper is to prove that in the space of compact operators on the space \(c_0\), \(K(c_0, c_0)\), for every natural number \(k\), there exists a \(k\)-dimensional subspace of \(K(c_0,c_0)\), which is not interpolating, but Chebyshev.