Shapiroʼs Theorem for subspaces

In the previous paper (Almira and Oikhberg, 2010 [4]), 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 ∈ A n ‖ x − a n ‖ ) decay arbitrarily slowly. In this work, we 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.