Some properties of Banach-valued sequence spaces \(\ell_p[X]\) (Q5957221)

Let \(X\) be a Banach space and \(X^*\) be its topological dual. 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 \(X\). For \(1\leq p<\infty\), let us consider the Banach-valued sequence space \(l_p[X]\), the space of weakly \(p\)-summable sequences on \(X\): \(l_p[X]=\{\overline{x}=(x_i)_i\in X^N: \sum_{i\geq 1} |f(x_i)|^p<\infty\) \(\forall f\in X^*\}\) and introduce the norm \(\|\cdot\|_p\) on \(l_p[X]\): NEWLINE\[NEWLINE\|\overline{x}\|_p=\sup \Biggl\{\Biggl(\sum_{i\geq 1} |f(x_i)|^p\Biggr)^{1/p}: f\in B_{X^*}\Biggr\},NEWLINE\]NEWLINE with respect to which \(l_p[X]\) is a Banach space. \(l_p[X]\) is called a GAK-space if for all \(\overline{x}\in l_p[X]\), \(\lim_n\|\overline{x}(i>n)\|_p=0\), where \(\overline{x}(i>n)=(0,\ldots,0,x_{n+1},x_{n+2},\ldots)\). The Köthe dual of \(l_p[X]\) with respect to the dual pair \((X,X^*)\) is denoted simply by \(l_p[X]^\times\), that is, NEWLINE\[NEWLINEl_p[X]^\times= \Biggl\{\overline{f}=(f_i)_i\in {X^*}^N: \quad \sum_{i\geq 1} |f_i(x_i)|^p<\infty \quad \forall \overline{x}\in l_p[X] \Biggr\}.NEWLINE\]NEWLINE It is noted that in general \(l_p[X]^\times\subset l_p[X]^*\). For \(1<q\leq\infty\), a Banach space \(X\) is said to have the \((q)\)-property if the following two statements about a sequence \((x_i)\) in \(X\) are equivalent: (i) \(\sum_{i\geq 1} t_ix_i\) converges for each \((t_i)\in l_q\); (ii) \(\sum_{i\geq 1} t_ix_i\) converges uniformly for all \((t_i)\in B_{l_q}\). It is known that the space \(l_p[X]\;(1<p<\infty)\) is a \(GAK\)-space if and only if \(X\) has the \((q)\)-property, where \(1/p+1/q=1\) [see \textit{M. Gupta} and \textit{Q. Bu}, J. Anal. 2, 103-113 (1994; Zbl 0818.46023)]. One of the main results of the paper is the following theorem. NEWLINENEWLINENEWLINETheorem 3.3. For \(1\leq p<\infty\), the following hold: (i) \(l_p[X]^\times\) is a closed subspace of \(l_p[X]^*\); (ii) \(\|\overline{x}\|_p=\sup\{|\langle\overline{x},\overline{f}\rangle|: \overline{f}\in l_p[X]^\times, \|\overline{f}\|_p^*\leq 1 \}\) for each \(\overline{x}\in l_p[X]\), where \(\|\cdot\|_p^*\) is the dual norm of \(\|\cdot\|_p\) on the dual space of \(l_p[X]\); (iii) \(l_p[X]^\times=l_p[X]^*\) if and only if \(l_p[X]\) is a GAK-space. NEWLINENEWLINENEWLINEUsing this result, the authors characterize the reflexivity of \(l_p[X]\) in terms of reflexivity and \((q)\)-property of a Banach space \(X\): NEWLINENEWLINENEWLINETheorem 3.5. The Banach space \(l_p[X]\) \((1<p<\infty)\) is a reflexive space if and only if (i) \(X\) is a reflexive space and (ii) \(l_p[X]\) is a GAK-space. NEWLINENEWLINENEWLINETheorem 3.6. The Banach space \(l_p[X]\) is a reflexive space if and only if the Banach space \(X\) is both a reflexive space and has the \((q)\)-property \((1<p<\infty\), \(1/p+1/q=1)\). NEWLINENEWLINENEWLINEThis result shows that the characterization of the reflexivity of \(l_p[X]\) is different from the characterization of the reflexivity of the space \(l_p(X)\), which was examined earlier by \textit{I. E. Leonard} [J. Math. Anal. Appl. 54, 245-265 (1976; Zbl 0343.46010)]. NEWLINENEWLINENEWLINEBesides, convergent sequences of \(l_p[X]\) with respect to the norm topology and with respect to the weak topology are characterized. Furthermore, compact subsets of \(l_p[X]\) and relatively weakly sequentially compact subsets of \(l_p[X]\) are also characterized.