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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the multiplicative completion of certain basic sequences in $ L\sp{p},$ $ 1<p<\infty $


Author: Ben-Ami Braun
Journal: Trans. Amer. Math. Soc. 176 (1973), 499-508
MSC: Primary 46E30; Secondary 42A60
MathSciNet review: 0313777
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Abstract: Boas and Pollard proved that given any basis $ \{ {f_n}\} _{n = 1}^\infty $ for $ {L^2}(E)$ one can delete the first k basis elements and then find a bounded measurable function M such that $ \{ M{f_n}\} _{n = k + 1}^\infty $ is total in $ {L^2}(E)$, that is, the closure of the linear span of the set $ \{ M{f_n}:n \geq k + 1\} $ is $ {L^2}(E)$. We improve this result by weakening the hypothesis to accept bases of $ {L^p}(E),1 < p < \infty $, and strengthening the conclusion to read serially total, that is, given any $ f \in {L^2}(E)$ one can find a sequence of reals $ \{ {a_n}\} _{n = k + 1}^\infty $ such that $ \Sigma _{n = k + 1}^\infty {a_n}M{f_n}$ converges to f in the norm. We also show that certain infinite deletions are possible.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0313777-9
PII: S 0002-9947(1973)0313777-9
Keywords: Schauder basis, multiplicative completion
Article copyright: © Copyright 1973 American Mathematical Society