SWhat changed
Banach proves that a bounded real-linear scalar functional on a subspace of a real normed space extends to the whole space with the same norm, and derives separation and closure criteria.
Scientific scope
Banach proves that a bounded real-linear scalar functional on a subspace of a real normed space extends to the whole space with the same norm, and derives separation and closure criteria. The assessed result extends a bounded real-linear functional on a subspace of a real normed space while preserving its norm. Completeness, separability and a closed-subspace assumption are not required. The selected 1929 paper is distinct from its sequel and later complex/vector-valued extensions. Hahn’s earlier contribution and the other inherited tools retain credit. WRS has not independently reconstructed every original proof step.
Evidence
Source-linked, scoped account of the original work. The published scope states the scientific assumptions and remaining verification limits. WRS has not independently certified the full proof, experimental result or system benchmark.
Contributors
- Stefan Banach · SCIENCE: Original source-listed scientific author of this scoped work; byline does not determine effort fractions.
- S. Banach, Sur les fonctionnelles linéaires, Studia Mathematica 1 (1929), 211–216
- IMPAN publisher notice for Banach 1929 functional-extension article
- Hans Hahn, Über lineare Gleichungssysteme in linearen Räumen, publisher bibliographic record (1927)
Provisional WRS assessment, edition 2026-10-16.1. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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