Shapiroʼs Theorem for subspaces

AbstractIn 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 (An) (defined by E(x,An)=infa∈An‖x−an‖) 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

Dependence of Kolmogorov Widths on the Ambient Space

We study the dependence of the Kolmogorov widths of a compact set on the ambient Banach space.Статья посвящена исследованию вопроса о том, насколько сильно могут изменяться колмогоровские поперечники при расширении банахова пространства, содержащего заданный компакт

The pp-Daugavet property for function spaces

A natural extension of the Daugavet property for $p$-convex Banach function spaces and related classes is analysed. As an application, we extend the arguments given in the setting of the Daugavet property to show that no reflexive space falls into this class