AWhat changed
A bounded sequence of functions need not approach one function, even if none of its terms grows without limit. Banach and Saks show that in Lp(0,1), for a finite exponent p greater than one, one can select a subsequence whose ordinary arithmetic averages do converge in norm. Their argument first uses an earlier compactness result of F. Riesz to obtain a weakly convergent subsequence, then carefully selects later terms to control partial-sum growth after the weak limit is subtracted. A step-function construction transfers the result to the corresponding space of summable numerical sequences.
Scientific scope
A bounded sequence of functions need not approach one function, even if none of its terms grows without limit. Banach and Saks show that in Lp(0,1), for a finite exponent p greater than one, one can select a subsequence whose ordinary arithmetic averages do converge in norm. Their argument first uses an earlier compactness result of F. Riesz to obtain a weakly convergent subsequence, then carefully selects later terms to control partial-sum growth after the weak limit is subtracted. A step-function construction transfers the result to the corresponding space of summable numerical sequences. The averages have equal weights along one subsequence; arbitrary convex combinations are not substituted. The result does not apply to every Banach space, L-infinity or every chosen subsequence. The collected reprint explicitly corrects the authors' claim that their L1 counterexample is weakly null: its functions are not uniformly integrable. That false weak-convergence claim is excluded. Failure for general bounded L1 sequences is distinct from the later theorem about weakly convergent L1 sequences. The linked collected scan is currently access-restricted; its first journal edition has not been collated here.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The averages have equal weights along one subsequence; arbitrary convex combinations are not substituted. The result does not apply to every Banach space, L-infinity or every chosen subsequence. The collected reprint explicitly corrects the authors' claim that their L1 counterexample is weakly null: its functions are not uniformly integrable. That false weak-convergence claim is excluded. Failure for general bounded L1 sequences is distinct from the later theorem about weakly convergent L1 sequences. The linked collected scan is currently access-restricted; its first journal edition has not been collated here.
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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