A permanence theorem for sums of sequence spaces
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- by A. K. Snyder PDF
- Proc. Amer. Math. Soc. 93 (1985), 489-492 Request permission
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.References
- G. Bennett, A new class of sequence spaces with applications in summability theory, J. Reine Angew. Math. 266 (1974), 49–75. MR 344846, DOI 10.1515/crll.1974.266.49
- A. K. Snyder, Universal families for conull FK spaces, Trans. Amer. Math. Soc. 284 (1984), no. 1, 389–399. MR 742431, DOI 10.1090/S0002-9947-1984-0742431-1
- Albert Wilansky, Topics in functional analysis, Lecture Notes in Mathematics, No. 45, Springer-Verlag, Berlin-New York, 1967. Notes by W. D. Laverell. MR 0223854
- Albert Wilansky, Summability through functional analysis, North-Holland Mathematics Studies, vol. 85, North-Holland Publishing Co., Amsterdam, 1984. Notas de Matemática [Mathematical Notes], 91. MR 738632
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 93 (1985), 489-492
- MSC: Primary 46A45; Secondary 40H05
- DOI: https://doi.org/10.1090/S0002-9939-1985-0774008-2
- MathSciNet review: 774008