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/