Semi-embeddings and weakly sequential completeness of the projective tensor product

Qingying Bu¹

¹ Department of Mathematics University of Mississippi University, MS 38677, U.S.A.

Studia Mathematica, Tome 169 (2005) no. 3, pp. 287-294

Résumé

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.

Détail

DOI : 10.4064/sm169-3-4

Keywords: boundedly complete unconditional schauder decomposition banach space weakly sequentially complete whenever weakly sequentially complete each mathbb through semi embeddings proof lewiss result banach spaces has unconditional basis mathbin widehat otimes projective tensor product weakly sequentially complete whenever weakly sequentially complete