For a Tychonoff space X, let \(C_k(X)\) and \(C_p(X)\) be the spaces of real-valued continuous functions C(X) on X endowed with the compact-open topology and the pointwise topology, respectively. If X is compact, the classic result of A. Grothendieck states that \(C_k(X)\) has the Dunford–Pettis property and the sequential Dunford–Pettis property. We extend Grothendieck’s result by showing that \(C_k(X)\) has both the Dunford–Pettis property and the sequential Dunford–Pettis property if X satisfies one of the following conditions: (1) X is a hemicompact space, (2) X is a cosmic space (= a continuous image of a separable metrizable space), (3) X is the ordinal space \([0,\kappa )\) for some ordinal \(\kappa \), or (4) X is a locally compact paracompact space. We show that if X is a cosmic space, then \(C_k(X)\) has the Grothendieck property if and only if every functionally bounded subset of X is finite. We prove that \(C_p(X)\) has the Dunford–Pettis property and the sequential Dunford–Pettis property for every Tychonoff space X, and \(C_p(X) \) has the Grothendieck property if and only if every functionally bounded subset of X is finite.

中文翻译：

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函数空间的Dunford–Pettis类型属性和Grothendieck属性

对于吉洪诺夫空间X，让\（C_K（X）\）和\（C_P（X）\）是实值连续函数的空间Ç（X）上X赋有紧致开拓扑和逐点拓扑， 分别。如果X是紧凑的，则A. Grothendieck的经典结果表明\（C_k（X）\）具有Dunford–Pettis属性和顺序的Dunford–Pettis属性。通过证明\（C_k（X）\）同时具有Dunford–Pettis属性和连续的Dunford–Pettis属性（如果X满足以下条件之一），可以扩展Grothendieck的结果：（1）X是半紧空间，（2）X是宇宙空间（=可分离的可量化空间的连续图像），（3）X是某些序数\（\ kappa的序数空间\（[0，\ kappa} \）\）或（4）X是局部紧凑的超紧空间。我们证明，如果X是一个宇宙空间，则\（C_k（X）\）具有且仅当X的每个函数有界子集都是有限的时才具有Grothendieck属性。我们证明\（C_p（X）\）具有每个Tychonoff空间X的Dunford–Pettis属性和顺序的Dunford–Pettis属性，并且\（C_p（X）\）当且仅当X的每个函数有界子集都是有限的时，才具有Grothendieck属性。