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Köthe dual of Banach sequence spaces \(\ell_ p [X] (1\leq p<\infty )\) and Grothendieck space. (English) Zbl 0785.46009
Let \(X\) be a Banach space and \(X^*\) its topological dual. For \(1\leq p<\infty\), let \(\ell_ p[X]= \{(x_ i)\): \(\sum_{i\geq 1} | f(x_ i)|^ p<\infty\) \(\forall f\in X^*\}\). Then \(\ell_ p[X]\) is a Banach space for each fixed \(p\), with the norm \[ \|(x_ i)\|_{(\ell_ p)}= \sup\Biggl\{\Biggl(\sum_{i\geq 1} | f(x_ i)|^ p\Biggr)^{1/p}:\;f\in B_{X^*}\Biggr\} \] where \(B_{X^*}\) denotes the closed unit ball of \(X^*\). The Köthe dual is defined as \[ \ell_ p[X]^ \times= \biggl\{(f_ i)\in X^*:\;\sum_{i\geq 1}| f_ i(x_ i)| <\infty\;\forall(x_ i)\text{ in }\ell_ p[X]\biggr\}. \] The authors establish the following result:
Let \(1\leq p<\infty\) and \({1\over p}+ {1\over q}=1\). Then \((f_ i)\in\ell_ p[X]^ \times \Leftrightarrow f_ i=\sum_{n\geq 1} r_ n s_ i^{(n)}h_ n\) (\(i=1,2,\dots)\) for some \((r_ n)\in\ell_ 1\), a bounded sequence \(\{s^{(n)}\}\) of sequences of \(\ell_ q\) and a bounded sequence \((h_ n)\) of \(X^*\).
There is also a necessary and sufficient condition in order that for each fixed \(p\) with \(1<p<\infty\), \(\ell_ p[X]\) is a Grothendieck space.
MSC:
46A45 Sequence spaces (including Köthe sequence spaces)
46B45 Banach sequence spaces
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