It is proved that if a Banach space $Y$ is a quotient of a Banach space having a shrinking unconditional basis, then every normalized weakly null sequence in $Y$ has an unconditional subsequence. The proof yields the corollary that every quotient of Schreier's space is $c_o$-saturated.
Smooth norms and approximation in Banach spaces of the type C(K)
Vector-valued Walsh-Paley martingales and geometry of Banach spaces
Embedding $\ell_{\infty}$ into the space of all Operators on Certain Banach Spaces
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