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Abstract

We show that if $\{P_k\}$ is a boundedly complete, unconditional Schauder decomposition of a Banach space $X$, then $X$ is weakly sequentially complete whenever $P_kX$ is weakly sequentially complete for each $k \in \mathbb N$. Then through semi-embeddings, we give a new proof of Lewis's result: if one of Banach spaces $X$ and $Y$ has an unconditional basis, then $X\mathbin{\widehat{\otimes}}Y$, the projective tensor product of $X$ and $Y$, is weakly sequentially complete whenever both $X$ and $Y$ are weakly sequentially complete.