Abstract
The skewness of Banach spaces was introduced by Fitzpatrick–Reznick. In this paper, we compute the skewness \(s(\ell _p-\ell _1)\) of Day–James spaces \(\ell _p-\ell _1\), where \(1< p < \infty\). This gives that the inequality \(s(X)\le 2 \rho _X(1)\) is strict for such space X, where \(\rho _X\) is the modulus of smoothness of X.
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References
Alonso, J., Llorens-Fuster, E.: Geometric mean and triangles inscribed in a semicircle in Banach spaces. J. Math. Anal. Appl. 340, 1271–1283 (2008)
Baronti, M., Papini, P.L.: Projections, skewness and related constants in real normed spaces. Math. Pannon. 3, 31–47 (1992)
Bonsall, F.F., Duncan, J.: Numerical Ranges II. London Math. Soc., Lecture Note Series 10. Cambridge University Press, Cambridge (1973)
Fitzpatrick, S., Reznick, B.: Skewness in Banach spaces. Trans. Am. Math. Soc. 275, 587–597 (1983)
Kato, M., Maligranda, L., Takahashi, Y.: On James and Jordan-von Neumann constants and the normal structure coefficient of Banach spaces. Stud. Math. 144, 275–295 (2001)
Mitani, K.-I., Saito, K.-S.: A note on relations between skewness and geometrical constants of Banach spaces. Linear Nonlinear Anal. 7, 257–264 (2021)
Mitani, K.-I., Saito, K.-S., Suzuki, T.: Smoothness of absolute norms on \({\mathbb{C} }^n\). J. Convex Anal. 10, 89–107 (2003)
Mitani, K.-I., Saito, K.-S., Takahashi, Y.: Skewness and James constant of Banach spaces. J. Nonlinear Convex Anal. 14, 115–122 (2013)
Mitani, K.-I., Saito, K.-S., Komuro, N.: Extremal structure of absolute norms and the skewness. Linear Nonlinear Anal. 1, 159–167 (2015)
Ritt, R.K.: A generalization of inner product. Mich. Math. J. 3, 23–26 (1955)
Takahashi, Y.: Some geometric constants of Banach spaces—A unified approach, Banach and function spaces II, pp. 191–220. Yokohama Publisher, Yokohama (2007)
Yang, C., Li, H.: On the James type constant of \(\ell _p-\ell _1\). J. Inequal. Appl. 2015, 79 (2015)
Acknowledgements
The first author was supported in part by Grants-in-Aid for Scientific Research (No. 21K03275), Japan Society for the Promotion of Science.
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Communicated by M. S. Moslehian.
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Mitani, KI., Saito, KS. & Komuro, N. Skewness of Day–James spaces. Ann. Funct. Anal. 13, 75 (2022). https://doi.org/10.1007/s43034-022-00222-4
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DOI: https://doi.org/10.1007/s43034-022-00222-4