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Evaluation maps, restriction maps, and compactness. (English) Zbl 0948.46008
If $B(K)$ 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 $B(K)$ is defined by: $E(x^*)(k)= \langle x^*,k\rangle$ 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 $B(K)$ 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).

46B20Geometry and structure of normed linear spaces
46B50Compactness in Banach (or normed) spaces
Full Text: EuDML