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For a Tychonoff space X , let $$C_k(X)$$ C k ( X ) and $$C_p(X)$$ 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 $$C_k(X)$$ C k ( X ) has the Dunford–Pettis property and the sequential Dunford–Pettis property. We extend Grothendieck’s result by showing that $$C_k(X)$$ 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 $$[0,\kappa )$$ [ 0 , κ ) for some ordinal $$\kappa $$ κ , or (4) X is a locally compact paracompact space. We show that if X is a cosmic space, then $$C_k(X)$$ C k ( X ) has the Grothendieck property if and only if every functionally bounded subset of X is finite. We prove that $$C_p(X)$$ C p ( X ) has the Dunford–Pettis property and the sequential Dunford–Pettis property for every Tychonoff space X , and $$C_p(X) $$ C p ( X ) has the Grothendieck property if and only if every functionally bounded subset of X is finite.

Dunford–Pettis type properties and the Grothendieck property for function spaces