If denotes the space of all bounded, real-valued functions on a bounded subset of a Banach space , the evaluation map from to is defined by: for all and . If denotes a bounded subset of instead of , analogous evaluation maps from and to are similarly defined.
In this paper, various properties of the set which are related to compactness in some way or another (e.g. weak compactness, the Dunford-Pettis property) are characterized in terms of corresponding properties of these evaluation maps or their restrictions to subspaces. The paper brings together in a unified fashion numerious results of this nature which are scattered through the literature (and proved by widely different techniques).