What is recorded here?
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. Suppose a linear measurement is understood only on a small part of a larger space, and its size is bounded by the size of its input. Can it be extended to all inputs without making that bound worse? Banach answers yes for real normed spaces. He first chooses a compatible value on one new direction, then extends through a well-ordering. The result also constructs measurements that distinguish a point from a subspace when their distance is positive. The space need not be complete or separable. This is an existence and structure result: it neither selects a unique extension nor supplies a general numerical algorithm. Earlier related work by Riesz, Helly and Hahn remains part of the attribution context.
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Scientific scope
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.
Read the original work
- S. Banach, Sur les fonctionnelles linéaires, Studia Mathematica 1 (1929), 211–216
- IMPAN publisher notice for Banach1929 functional-extension article
- Hans Hahn, Über lineare Gleichungssysteme in linearen Räumen, publisher bibliographic record (1927)
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People and systems
Stefan Banach
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English catalogue narrative
Banach: extending a bounded linear measurement without increasing its norm
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.
Suppose a linear measurement is understood only on a small part of a larger space, and its size is bounded by the size of its input. Can it be extended to all inputs without making that bound worse? Banach answers yes for real normed spaces. He first chooses a compatible value on one new direction, then extends through a well-ordering. The result also constructs measurements that distinguish a point from a subspace when their distance is positive. The space need not be complete or separable. This is an existence and structure result: it neither selects a unique extension nor supplies a general numerical algorithm. Earlier related work by Riesz, Helly and Hahn remains part of the attribution context.
Scientific scope
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.
People and systems
Stefan Banach