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Banach-Mackey, locally complete spaces, and $$\ell_{p,q}$$-summability. (English) Zbl 0961.46001
A $$p$$-absolutely summable sequence $$(x_n)\subset E$$ in a locally convex space $$E$$ with $$1\leq p\leq \infty$$ is said to be $$\ell_{p,q}$$-summable if for every $$(\lambda_n)\in \ell_q$$, the series $$\sum^\infty_{n= 1}\lambda_n x_n$$ converges to $$x$$ for some $$x\in E$$. The space $$E$$ has the $$\ell_{p,q}$$-summability property if each $$p$$-absolutely summable sequence is $$\ell_{p,q}$$-summable. The authors prove that if $$E$$ has this property for $$1\leq p, q\leq\infty$$ with $$1/p+ 1/q= 1$$, then the space $$E$$ is locally complete.
##### MSC:
 46A03 General theory of locally convex spaces 46A17 Bornologies and related structures; Mackey convergence, etc.
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