Sliding hump technique and spaces with the Wilansky property
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- by Dominikus Noll and Wolfgang Stadler PDF
- Proc. Amer. Math. Soc. 105 (1989), 903-910 Request permission
Abstract:
We prove that if $E$ is a BK- AK-space whose dual $E’$ as well is BK- AK , then $\sigma (E’,F)$ and $\sigma (E’,F)$ have the same convergent sequences whenever $F$ is a subspace of $E''$ containing $\Phi$ and satisfying ${F^\beta } = {E^\beta }$. This extends a result due to Bennett [B$_{2}$] and the second author [S]. We provide new examples of BK-spaces having the Wilansky property. We show that the bidual $E''$ of a solid BK- AK-space $E$ whose dual as well is BK- AK satisfies a separable version of the Wilansky property. This extends a theorem of Bennett and Kalton, who proved that ${l^\infty }$ has the separable Wilansky property.References
- G. Bennett, Some inclusion theorems for sequence spaces, Pacific J. Math. 46 (1973), 17–30. MR 331007
- Grahame Bennett, Sequence spaces with small $\beta$-duals, Math. Z. 194 (1987), no. 3, 321–329. MR 879935, DOI 10.1007/BF01162240
- G. Bennett and N. J. Kalton, Addendum to: “$FK$-spaces containing $c_{0}$”, Duke Math. J. 39 (1972), 819–821. MR 313758
- G. Bennett and N. J. Kalton, Inclusion theorems for $K$-spaces, Canadian J. Math. 25 (1973), 511–524. MR 322474, DOI 10.4153/CJM-1973-052-2
- S. Kwapień, On Banach spaces containing $c_{0}$, Studia Math. 52 (1974), 187–188. MR 356156
- Dominikus Noll and Wolfgang Stadler, Zerlegungen von Wachstumsbereichen und Wirkfeldern für die Verfahren bewichteter Mittel, Manuscripta Math. 60 (1988), no. 2, 197–209 (German, with English summary). MR 924087, DOI 10.1007/BF01161929
- A. Pełczyński, A note on the paper of I. Singer “Basic sequences and reflexivity of Banach spaces”, Studia Math. 21 (1961/62), 371–374. MR 146636, DOI 10.4064/sm-21-3-370-374
- A. K. Snyder, A property of the embedding of $c_0$ in $l_\infty$, Proc. Amer. Math. Soc. 97 (1986), no. 1, 59–60. MR 831387, DOI 10.1090/S0002-9939-1986-0831387-6
- Wolfgang Stadler, Zu einer Frage von Wilansky, Arch. Math. (Basel) 48 (1987), no. 2, 149–152 (German). MR 878426, DOI 10.1007/BF01189285
- 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 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 105 (1989), 903-910
- MSC: Primary 46A45; Secondary 40D25, 46A07
- DOI: https://doi.org/10.1090/S0002-9939-1989-0989099-7
- MathSciNet review: 989099