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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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