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Suppose we have two Banach spaces X and Y each of them having the Schur property (weakly convergent sequences are norm convergent). Does it follows that X+Y has the Schur property? Note that this is trivially true when the sum is direct.

Any proof (or disproof), or references will be appreciated.

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    $\begingroup$ Meaning that they are both isometrically embedded in some Banach space $Z$ and you consider the closure of the linear span as the new space, or something else? $\endgroup$
    – fedja
    Commented Dec 6, 2017 at 20:57
  • $\begingroup$ @fedja. We can say that the pair (X,Y) is a compatible couple in the sense of interpolation theory: there is a Hausdorff topological vector space Z such that X and Y are subspaces of Z. The space X+Y is considered with the usual norm and is complete since both X,Y are complete. (to be more clear I am using the definition from p. 25 in Bergh, Lofstrom) $\endgroup$
    – Eduard
    Commented Dec 7, 2017 at 14:13

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