Banach spaces which are Grothendieck spaces with the Dunford-Pettis property (briefly, GDP) are classical. A systematic treatment of GDP-Fréchet 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-spaces 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 $K_p(\ov{V})$ are GDP-spaces if and only if they are Montel. On the other hand, $K_\infty(\ov{V})$ is always a GDP-space and $k_\infty(V)$ is a GDP-space whenever its (Fréchet) predual, i.e., the Kothe echelon space $\lambda_1(A)$, is distinguished.
Grothendieck spaces with the Dunford-Pettis property
ALBANESE, Angela Anna;
2010-01-01
Abstract
Banach spaces which are Grothendieck spaces with the Dunford-Pettis property (briefly, GDP) are classical. A systematic treatment of GDP-Fréchet 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-spaces 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 $K_p(\ov{V})$ are GDP-spaces if and only if they are Montel. On the other hand, $K_\infty(\ov{V})$ is always a GDP-space and $k_\infty(V)$ is a GDP-space whenever its (Fréchet) predual, i.e., the Kothe echelon space $\lambda_1(A)$, is distinguished.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
