Basis entropy in Banach spaces

Abstract

We introduce and study two notions of entropy in a Banach space X with a normalized Schauder basis . The geometric entropy E(A) of a subset A of X is defined to be the infimum of radii of compact bricks containing A. We obtain several compactness characterizations for bricks (Theorem 3.7) useful for main results. We also obtain sufficient conditions on a set in a Hilbert space to have finite unconditional entropy. For Banach spaces without a Schauder basis we offer another entropy, called the Auerbach entropy. Finally, we pose some open problems.