## 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 , κ) for some ordinal κ, 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.

Original language | English |
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Pages (from-to) | 871-884 |

Number of pages | 14 |

Journal | Revista Matematica Complutense |

Volume | 33 |

Issue number | 3 |

DOIs | |

State | Published - 1 Sep 2020 |

## Keywords

- Dunford–Pettis property
- Function space
- Grothendieck property
- Sequential Dunford–Pettis property

## ASJC Scopus subject areas

- General Mathematics