Banach-Mackey, locally complete spaces, and \(\ell_{p,q}\)-summability (Q1578363)

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.