Shapiro's theorem for subspaces (Q662069)

From authors' abstract: In an earlier preprint [``Approximation schemes satisfying Shapiro's Theorem'', submitted, \url{arXiv:1003.3411}], the authors investigated the existence of an element \(x\) of a quasi-Banach space \(X\) whose errors of best approximation by a given approximation scheme \((A_n)\) (defined by \(E(x,A_n)=\inf_{a\in A_n}\| x-a_n\|\)) decay arbitrarily slowly. In this work, they consider the question of whether \(x\) witnessing the slowness rate of approximation can be selected in a prescribed subspace of \(X\). In many particular cases, the answer turns out to be positive.