Weakly sequential completeness of Banach-valued sequence space \(\ell_p[X]\) (Q2725633)

Let \((X,X^*)\) be a dual pair, where \(X\) is a Banach space and \(X^*\) is its topological dual. Let us denote the strong topology and the weak topology with respect to the dual pair \((X,X^*)\) by \(\beta(X,X^*)\) and \(\sigma(X,X^*)\) respectively. \(B(X)\) stands for the closed unit ball of \(X\) and \(X^N\) -- the Cartesian product of countably many copies of a Banach space \(X\). For \(1\leq p<\infty,\) let us introduce the weakly \(p\)-summable sequence space \(\ell_p[X]=\{\overline x=(x_i)\in X^N\): for each \(f\in X^*\), \(\sum_{i\geq 1}|f(x_i)|^p<\infty\}\) and introduce a norm \(\|\cdot\|_p\) on \(\ell_p[X]\): \(\|\overline x\|_p=\sup\{(\sum_{i\geq 1}|f(x_i)|^p)^{1/p}: f\in B(X^*)\}.\) For \(\overline x\in X^N\) we introduce the symbol \(\overline x(i>n)=(0,\ldots,0,x_{n+1},x_{n+2},\ldots).\) The Banach-valued sequence space \(\ell_p[X]\) is called a GAK-space if for each \(\overline x\in \ell_p[X]\), \(\lim_n\|\overline x(i>n)\|=0\). We denote the GAK-subspace of \(\ell_p[X]\) by \(\ell_p[X]_r=\{\overline x\in \ell_p[X]: \lim_n\|\overline x(i>n)\|=0\}.\) The Köthe dual of \(\ell_p[X]\) with respect to the dual pair \((X,X^*)\) is denoted by \(\ell_p[X]^\times= \{\overline f=(f_i)\in X^{*N}\): for each \(\overline x\in l_p[X]\), \(\sum_{i\geq 1}|f_i(x_i)<\infty|\}.\) For \(1<p\leq\infty\), a Banach space \(X\) is said to have the \((p)\)-property if the following two statements about a sequence \((x_i)\) in \(X\) are equivalent: NEWLINENEWLINENEWLINE(i) \(\sum_{i\geq 1}t_ix_i\) converges for each \((t_i)\in \ell_p\); NEWLINENEWLINENEWLINE(ii) \(\sum_{i\geq 1}t_ix_i\) converges uniformly for all \((t_i)\in B(\ell_p)\). NEWLINENEWLINENEWLINEThe main results of the paper are the following theorems. NEWLINENEWLINENEWLINETheorem 3.3. \(\ell_p[x]\) \((1\leq p<\infty)\) is \(\sigma(\ell_p[X], \ell_p[X]^\times)\)-sequentially complete if and only if \(X\) is \(\sigma(X,X^*)\)-sequentially complete. NEWLINENEWLINENEWLINETheorem 3.4. \(\ell_p[x]_r\) \((1\leq p<\infty)\) is \(\sigma(\ell_p[X]_r, (\ell_p[X]_r)^*)\)-sequentially complete if and only if NEWLINENEWLINENEWLINE(i) \(X\) is \(\sigma(X,X^*)\)-sequentially complete and NEWLINENEWLINENEWLINE(ii) \(\ell_p[X]\) is a GAK-space. NEWLINENEWLINENEWLINESome corollaries of these results are given. Let us note one of these corollaries. NEWLINENEWLINENEWLINECorollary 3.8. If \(X\) is \(\sigma(X,X^*)\)-sequentially complete space and has the \((q)\)-property, then the space \(\ell_p[X]\) is \(\sigma(\ell_p[X], (\ell_p[X])^*)\)-sequentially complete space \((1<p<\infty\), \(1/p+1/q=1)\).NEWLINENEWLINEFor the entire collection see [Zbl 0960.00019].