Estimates for Covering Numbers in Schauder’s Theorem about Adjoints of Compact Operators

Let T : X → Y be a bounded linear map between Banach spaces X and Y . Let T ∗ : Y ∗ → X∗ be its adjoint. Let BX and BY ∗ be the closed unit balls of X and Y ∗ respectively. We obtain apparently new estimates for the covering numbers of the set T ∗ (BY ∗ ). These are expressed in terms of the covering numbers of T (BX), or, more generally, in terms of the covering numbers of a “significant” subset of T (BX). The latter more general estimates are best possible. These estimates follow from our new quantitative version of an abstract compactness result which generalizes classical theorems of Arzelà-Ascoli and of Schauder. Analogous estimates also hold for the covering numbers of T (BX), in terms of the covering numbers of T ∗ (BY ∗) or in terms of a suitable “significant” subset of T ∗ (BY ∗ ).