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

Abstract

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.