AWhat changed
An infinite matrix can turn one sequence of numbers into another. Schur asks when it turns every bounded input into a convergent output. The answer requires the rows to settle coordinate by coordinate and their absolute tails to become uniformly small. Equivalently, the absolute row sums must approach the absolute sum of the limiting row. If the tails remain uncontrolled, his proof builds a bounded test sequence that makes the outputs oscillate. In modern sequence-space language, this gives the Schur property: a weakly convergent sequence of absolutely summable sequences also converges in its norm.
Scientific scope
An infinite matrix can turn one sequence of numbers into another. Schur asks when it turns every bounded input into a convergent output. The answer requires the rows to settle coordinate by coordinate and their absolute tails to become uniformly small. Equivalently, the absolute row sums must approach the absolute sum of the limiting row. If the tails remain uncontrolled, his proof builds a bounded test sequence that makes the outputs oscillate. In modern sequence-space language, this gives the Schur property: a weakly convergent sequence of absolutely summable sequences also converges in its norm. Coordinatewise convergence alone is insufficient. The modern l1 formulation comes from pairing the matrix rows with all bounded scalar tests. It concerns sequences, not equality of the weak and norm topologies, arbitrary nets or every Banach space. The entrywise-product and triangularization theorems bearing Schur's name are separate contributions. Toeplitz's regular summability criterion and Steinhaus's no-universal-regular-summation result retain their explicit predecessor credit.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. Coordinatewise convergence alone is insufficient. The modern l1 formulation comes from pairing the matrix rows with all bounded scalar tests. It concerns sequences, not equality of the weak and norm topologies, arbitrary nets or every Banach space. The entrywise-product and triangularization theorems bearing Schur's name are separate contributions. Toeplitz's regular summability criterion and Steinhaus's no-universal-regular-summation result retain their explicit predecessor credit.
Contributors
- The named authors in the research record · SCIENCE: Original-paper scientific authors for this exact mathematical contribution. Independent papers grouped in one historical family are not a joint byline. Earlier theorems, supporting proof contributions and later generalizations remain separate. Individual effort shares, author nationality and unresolved cross-record identities are not inferred.
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Provisional WRS assessment, edition 2026-10-23.7. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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