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.



Reference request for sums of Grothendieck spaces
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.


