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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

A permanence theorem for sums of sequence spaces


Author: A. K. Snyder
Journal: Proc. Amer. Math. Soc. 93 (1985), 489-492
MSC: Primary 46A45; Secondary 40H05
MathSciNet review: 774008
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Abstract: Let $ l$ be the space of absolutely summable sequences. Using difficult functional analytic techniques Bennett proved that if $ X$ is a separable FK space containing $ {\delta ^n}$ for all $ n$ and if $ {\delta ^n} \to 0$ in $ X + l$, then $ l \subset X$. Bennett also asked whether the separability assumption can be dropped. Using an elementary invertibility criterion for Banach algebras, the present note gives a self-contained proof that if $ z$ is a null sequence, $ X$ is an $ {\text{FK}}$ space containing $ {\delta ^n}$ for all $ n$, and $ X + zl = l$, then $ X = l$. This answers Bennett's question in the affirmative.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0774008-2
PII: S 0002-9939(1985)0774008-2
Keywords: FK space, wedge space
Article copyright: © Copyright 1985 American Mathematical Society