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How do I show that for any increasing sequence $(n_k) \subseteq \mathbb{N}$, the space $\left( \oplus _{k=1} ^\infty l_2 ^{(n_k)} \right) _\infty$ contains a complemented isometric copy of $l_2$?

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    $\begingroup$ "How do I show ...": why do you know your assertion is true? $\endgroup$
    – Stefan Kohl ♦
    Commented Mar 18, 2015 at 20:51
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    $\begingroup$ I think it is true because I am studying the paper "An alternative Dunford-Pettis property" by W. Freedman and he made this statement. Now I am trying to prove it. $\endgroup$
    – Mary
    Commented Mar 18, 2015 at 20:58

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See example 4 in On the Dunford-Pettis property in Banach spaces. J. M. F. Castillo, M. Gonzales

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See the second paragraph of my first answer to

Ultraweak topology on B(X): Is the map X\otimes X* -> B(X)* isometric?

It references my paper

[J] Johnson, William B. A complementary universal conjugate Banach space and its relation to the approximation problem. Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972). Israel J. Math. 13 (1972), 301–310 (1973),

although the particular case you want is due to Stegall (referenced in the above paper as well as in the Castillo-Gonzalez paper mentioned by Norbert).

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