Weakly sequential completeness of Banach-valued sequence spaces \(\ell_p[X]\) (Q5932742)

For \(1\leq p <\infty\), the Banach-valued sequence space \(\ell_p[X]\) has been introduced as the space of weakly \(p\)-summable sequences on \(X\). The Köthe dual of \(\ell_p[X]\) with respect to the dual pair \((X,X^*)\) has been denoted by \(\ell_p[X]^x\). With \(\sigma(X,X^*)\) as the weak topology with respect to the dual pair \((X,X^*)\), the author has proved that \(\ell_p[X]\) is \(\sigma(\ell_p[X],\ell_p[X]^x)\)-sequentially complete iff \(x\) is \(\sigma (X,X^*)\)-sequentially complete. A subspace of \(\ell_p[X]\) defined as \(\ell_p[X]_G=\{\bar x\in \ell_p[x]:\lim_n\|\bar x(i>n)\|_p=0\}\) has been termed as \(GAK\)-subspace of \(\ell_p[x]\). A characterization theorem for the weakly sequential completeness of the space \(\ell_p[x]_G\) has also been proved.