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A diagonal theorem. Answer to a question of Antosik. (English) Zbl 0799.40005
The author is able to remove a hypothesis in a theorem of P. Antosik to prove: If \((x_{ij})\), \(i,j \in \mathbb{N}\) is a matrix with values in a commutative topological group, and every increasing sequence \((m_ i)\) in \(\mathbb{N}\) has a subsequence \((n_ i)\) such that \(\lim_ i \sum^ \infty_{j = 1} x_{n_ i n_ j} = 0\), then \(\lim_ i x_{i,i} = 0\).

40J05 Summability in abstract structures
46A99 Topological linear spaces and related structures
40A05 Convergence and divergence of series and sequences
22A10 Analysis on general topological groups
40C05 Matrix methods for summability