w\(^*\)-basic sequences and reflexivity of Banach spaces. (English) Zbl 1081.46017

Summary: 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 not reflexive for some \(X_1\subset X\), \(X_1\) having a basis.


46B28 Spaces of operators; tensor products; approximation properties
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