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.
We characterize the reflexivity of the completed projective tensor products $X{\widetilde{\otimes }_\pi } Y$ of Banach spaces in terms of certain approximative biorthogonal systems.