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I much appreciate your help with my previous question. Now, I'd like to ask about a reference. I need a fact asserting that if $p\in (1,\infty]$ (with emphasis on the case $p=\infty$) then the $\ell_p$-sum of an arbitrary (possibly uncountable) family of Grothendieck spaces is Grothendieck.

Best wishes, A.

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up vote 2 down vote accepted

For $1< p<\infty$ it is an exercise (which, in fact, some of my students did this past semester).

For $p=\infty$ it is false. The space $(\ell_1^1\oplus \ell_1^2 \oplus \ell_1^3 \oplus \dots)_\infty$ contains a norm one complemented subspace that is isometrically isomorphic to $\ell_1$ (another exercise for my students this past term), and being a Grothendieck space passes to complemented subspaces.

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For $p\in (1,\infty)$ the proof (of this exercise) can be found in (Lemma 1). – Tomek Kania Sep 11 '14 at 16:06

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