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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 $P_{k}X$ is weakly sequentially complete for each k ∈ 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 ⊗̂ Y, the projective tensor product of X and Y, is weakly sequentially complete whenever both X and Y are weakly sequentially complete.

Let E be a Banach space with 1-unconditional basis. Denote by $\Delta \left(\otimes {̂}_{n,\pi }E\right)$ (resp. $\Delta \left(\otimes {̂}_{n,s,\pi }E\right)$) the main diagonal space of the n-fold full (resp. symmetric) projective Banach space tensor product, and denote by $\Delta \left(\otimes {̂}_{n,\left|\pi \right|}E\right)$ (resp. $\Delta \left(\otimes {̂}_{n,s,\left|\pi \right|}E\right)$) the main diagonal space of the n-fold full (resp. symmetric) projective Banach lattice tensor product. We show that these four main diagonal spaces are pairwise isometrically isomorphic, and in addition, that they are isometrically lattice isomorphic to $E_{\left[n\right]}$, the completion of the n-concavification of...