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Evaluation maps, restriction maps, and compactness. (English) Zbl 0948.46008

If $B\left(K\right)$ denotes the space of all bounded, real-valued functions on a bounded subset $K$ of a Banach space $X$, the evaluation map $E$ from ${X}^{*}$ to $B\left(K\right)$ is defined by: $E\left({x}^{*}\right)\left(k\right)=〈{x}^{*},k〉$ for all ${x}^{*}\in {X}^{*}$ and $k\in K$. If $K$ denotes a bounded subset of ${X}^{*}$ instead of $X$, analogous evaluation maps from $X$ and ${X}^{**}$ to $B\left(K\right)$ are similarly defined.

In this paper, various properties of the set $K$ 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).

##### MSC:
 46B20 Geometry and structure of normed linear spaces 46B50 Compactness in Banach (or normed) spaces