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

Abstract

For a Tychonoff space X, let C k (X) and Cp(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: (i) X is a hemicompact space, (ii) X is a cosmic space (=a continuous image of a separable metrizable space), (iii) X is the ordinal space [0, κ) for some ordinal κ, or (vi) 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 Cp(X) has the Dunford-Pettis property and the sequential Dunford-Pettis property for every Tychonoff space X, and Cp(X) has the Grothendieck property if and only if every functionally bounded subset of X is finite.2000 Mathematics Subject Classification. Primary 46A03; Secondary 54H11. Key words and phrases. function space, Dunford-Pettis property, Grothendieck property. Jerzy Kakol gratefully acknowledges the nancial support he received from the Center for Advanced Studies in Mathematics of the Ben Gurion University of the Negev during his visit April, 2018.

Grothendieck spaces: the landscape and perspectives

“…In the literature concerning Fréchet spaces, the Grothendieck property is often considered in tandem with the Dunford-Pettis property; spaces satisfying both properties are termed GDP spaces. Their systematic treatment may be found, e.g., in [2,3,16]; for non-Fréchet spaces see, e.g., [56].…”

Locally convex properties of free locally convex spaces

“…Recall that an lcs E is said to have the Grothendieck property if every weak- * convergent sequence in the strong dual E ′ β is weakly convergent. We proved in [15] that C p (X) has the Grothendieck property if and only if every functionally bounded subset of X is finite, and if X is a sequential space, then C k (X) has the Grothendieck property if and only if X is discrete.…”

“…Following Grothendieck [18], an lcs E is said to have the Dunford-Pettis property ((DP ) property for short) if every continuous linear operator T from E into a quasi-complete locally convex space F , which transforms bounded sets of E into relatively weakly compact subsets of F , also transforms absolutely convex weakly compact subsets of E into relatively compact subsets of F . In [15], we proved that C p (X) has the (DP ) property for every Tychonoff space X and showed that C k (X) has the (DP ) property if X is hemicompact or a cosmic space.…”

“…Extending this result to locally convex spaces and following [13], we consider the following "sequential" version of the (DP ) property: an lcs E is said to have the sequential Dunford-Pettis property ((sDP ) property) if given weakly null sequences {x n } n∈N and {χ n } n∈N in E and the strong dual E ′ β of E, respectively, then lim n χ n (x n ) = 0. In [15], we showed that C p (X) has the (sDP ) property for every Tychonoff space X. If X is an ordinal space or a locally compact paracompect space, then C k (X) has the (sDP ) property (see [15]).…”

“…In [15], we showed that C p (X) has the (sDP ) property for every Tychonoff space X. If X is an ordinal space or a locally compact paracompect space, then C k (X) has the (sDP ) property (see [15]).…”