In 1953, Grothendieck [G] characterized locally convex Hausdorff spaces which have the Dunford-Pettis property and used this property to characterize weakly compact operators u : C(K) → F , where K is a compact Hausdorff space and F is a locally convex Hausdorff space (briefly, lcHs) which is complete. Among other results, he also showed that there is a bijective correspondence between the family of all F -valued weakly compact operators u on C(K) and that of all F -valued σ-additive Baire measures on K. But he did not develop any theory of integration to represent these operators.

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前言

1953 年，Grothendieck [G] 刻画了具有 Dunford-Pettis 性质的局部凸 Hausdorff 空间，并使用该性质刻画了弱紧算子 u : C(K) → F ，其中 K 是紧致 Hausdorff 空间，F 是局部凸完整的 Hausdorff 空间（简称为 lcHs）。在其他结果中，他还表明 C(K) 上所有 F 值弱紧算子 u 的族与 K 上所有 F 值 σ-可加贝尔测度的族之间存在双射对应关系。但他没有发展任何表示这些算子的积分理论。