We show that if is a boundedly complete, unconditional Schauder decomposition of a Banach space X, then X is weakly sequentially complete whenever 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm169-3-4, author = {Qingying Bu}, title = {Semi-embeddings and weakly sequential completeness of the projective tensor product}, journal = {Studia Mathematica}, volume = {166}, year = {2005}, pages = {287-294}, zbl = {1093.46043}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm169-3-4} }
Qingying Bu. Semi-embeddings and weakly sequential completeness of the projective tensor product. Studia Mathematica, Tome 166 (2005) pp. 287-294. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm169-3-4/