A permanence theorem for sums of sequence spaces
A. K. Snyder
Proc. Amer. Math. Soc. 93 (1985), 489-492
Primary 46A45; Secondary 40H05
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Abstract: Let be the space of absolutely summable sequences. Using difficult functional analytic techniques Bennett proved that if is a separable FK space containing for all and if in , then . 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 is a null sequence, is an space containing for all , and , then . This answers Bennett's question in the affirmative.
Bennett, A new class of sequence spaces with applications in
summability theory, J. Reine Angew. Math. 266 (1974),
0344846 (49 #9585)
K. Snyder, Universal families for conull FK
spaces, Trans. Amer. Math. Soc.
284 (1984), no. 1,
742431 (86a:46009), http://dx.doi.org/10.1090/S0002-9947-1984-0742431-1
Wilansky, Topics in functional analysis, Notes by W. D.
Laverell. Lecture Notes in Mathematics, No. 45, Springer-Verlag, Berlin,
0223854 (36 #6901)
Wilansky, Summability through functional analysis,
North-Holland Mathematics Studies, vol. 85, North-Holland Publishing
Co., Amsterdam, 1984. Notas de Matemática [Mathematical Notes], 91.
- G. Bennett, A new class of sequence spaces with applications in summability theory, J. Reine Angew. Math. 266 (1974), 49-75. MR 0344846 (49:9585)
- A. K. Snyder, Universal families for conull FK spaces, Trans. Amer. Math. Soc. 284 (1984), 389-399. MR 742431 (86a:46009)
- A. Wilansky, Topics in functional analysis, Lecture Notes in Math., Vol. 45, Springer-Verlag, 1967. MR 0223854 (36:6901)
- -, Summability through functional analysis, North-Holland, Amsterdam, 1984. MR 738632 (85d:40006)
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