Geodesic
$w^*$-basic sequences and reflexivity of Banach spaces
Czechoslovak Mathematical Journal, Tome 55 (2005) no. 3, pp. 677-681.

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We observe that a separable Banach space $X$ is reflexive iff each of its quotients with Schauder basis is reflexive. Similarly if $\mathcal L(X,Y)$ is not reflexive for reflexive $X$ and $Y$ then $\mathcal L(X_1, Y)$ is is not reflexive for some $X_1\subset X$, $X_1$ having a basis.
Classification : 46B10, 46B15, 46B28
Keywords: reflexive Banach space; Schauder basis; quotient space; w$^*$-basic sequence; tensor product 
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John, Kamil. $w^*$-basic sequences and reflexivity of Banach spaces. Czechoslovak Mathematical Journal, Tome 55 (2005) no. 3, pp. 677-681. http://geodesic.mathdoc.fr/item/CMJ_2005__55_3_a8/