Entropy of C(K)-Valued Operators Original Research Article

We investigate how the entropy numbers (en(T)) of an arbitrary Hölder-continuous operator T: E→C(K) are influenced by the entropy numbers (εn(K)) of the underlying compact metric space K and the geometry of E. We derive diverse universal inequalities relating finitely many εn(K)ʹs with finitely many en(T)ʹs which yield statements about the asymptotically optimal behaviour of the sequence (en(T)) in terms of the sequence (εn(K)). As an application we present new methods for estimating the entropy numbers of a precompact and convex subset in a Banach space E, provided that the entropy numbers of its extremal points are known.