On convergence of sections of sequences in Banach spaces
Let \(X\) be a Banach space, \(X^*\) be its dual space and let \(\ell^p,1\leq p<\infty\), be the Banach space of \(p\)-absolutely summable scalar sequences. For any sequence \((x_i)\) in \(X\) we set \[ (x_i)(\geq n)=(0,0,\ldots,0,x_n,x_{n+1},\ldots). \] Let us consider the Banach space \[ \ell_w^p(X)=\{(x_i)\subset X: (a(x_i))\in \ell^p, \forall a\in X^*\} \] with the norm \[ \varepsilon_p((x_i))=\sup\limits_{\|a\|\leq 1}\|(a(x_i))\|_{l^p} \] and let us denote \[ \ell_c^p(X)=\{(x_i)\in \ell_w^p(X): \varepsilon_p((x_i)(\geq n))\to 0 \;\;{\text{if}} \;\;n\to\infty\}. \] In the paper an elementary proof of the following known fact is given: Proposition. \(\ell_w^p(X)=l_c^p(X)\) (for \(1\leq p<\infty\)) if and only if \(l_w^p(X)\subset c_0\{X\}\), where \(c_0\{X\}\) is the space of norm null sequences in \(X\). A similar result for a family of Orlicz sequence spaces is proved too. Some applications to spaces of compact operators on Banach sequence spaces are considered.
scientific article; zbMATH DE number 1462211