Grothendieck spaces with the Dunford–Pettis property

Abstract

Banach spaces which are Grothendieck spaces with the Dunford–Pettis property (briefly, GDP) are classical. A systematic treatment of GDP-Frechet spaces occurs in Bonet and Ricker (Positivity 11:77–93, 2007). This investigation is continued here for locally convex Hausdorff spaces. The product and (most) inductive limits of GDP-space are again GDP-spaces. Also, every complete injective space is a GDP-space. For $${p\in \{0\}\cup[1,\infty)}$$ it is shown that the classical co-echelon spaces k p (V) and $${K_p(\overline{V})}$$ are GDP-spaces if and only if they are Montel. On the other hand, $${K_\infty(\overline{V})}$$ is always a GDP-space and k ∞(V) is a GDP-space whenever its (Frechet) predual, i.e., the Kothe echelon space λ 1(A), is distinguished.