# Grothendieck spaces with the Dunford–Pettis property

Grothendieck spaces with the Dunford–Pettis property 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-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 (Fréchet) predual, i.e., the Köthe echelon space λ 1(A), is distinguished. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Grothendieck spaces with the Dunford–Pettis property

, Volume 14 (1) – Apr 17, 2009
20 pages

/lp/springer_journal/grothendieck-spaces-with-the-dunford-pettis-property-Na7PhjLlkx
Publisher
Springer Journals
Subject
Mathematics; Econometrics; Calculus of Variations and Optimal Control; Optimization; Potential Theory; Operator Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-009-0011-x
Publisher site
See Article on Publisher Site

### 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-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 (Fréchet) predual, i.e., the Köthe echelon space λ 1(A), is distinguished.

### Journal

PositivitySpringer Journals

Published: Apr 17, 2009

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