On octahedrality and Müntz spaces
DOI:
https://doi.org/10.7146/math.scand.a-119844Abstract
We show that every Müntz space can be written as a direct sum of Banach spaces $X$ and $Y$, where $Y$ is almost isometric to a subspace of $c$ and $X$ is finite dimensional. We apply this to show that no Müntz space is locally octahedral or almost square.
References
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Werner, D., A remark about Müntz spaces, http://page.mi.fu-berlin.de/werner99/preprints/muentz.pdf, 2008.
Abrahamsen, T. A., Langemets, J., and Lima, V., Almost square Banach spaces, J. Math. Anal. Appl. 434 (2016), no. 2, 1549–1565. https://doi.org/10.1016/j.jmaa.2015.09.060
Abrahamsen, T. A., Leraand, A., Martiny, A., and Nygaard, O., Two properties of Müntz spaces, Demonstr. Math. 50 (2017), no. 1, 239–244. https://doi.org/10.1515/dema-2017-0025
Albiac, F. and Kalton, N. J., Topics in Banach space theory, second ed., Graduate Texts in Mathematics, vol. 233, Springer, 2016. https://doi.org/10.1007/978-3-319-31557-7
Becerra Guerrero, J., López-Pérez, G., and Rueda Zoca, A., Some results on almost square Banach spaces, J. Math. Anal. Appl. 438 (2016), no. 2, 1030–1040. https://doi.org/10.1016/j.jmaa.2016.02.041
Borwein, P. and Erdélyi, T., Generalizations of Müntz's theorem via a Remez-type inequality for Müntz spaces, J. Amer. Math. Soc. 10 (1997), no. 2, 327–349. https://doi.org/10.1090/S0894-0347-97-00225-7
Bourgin, R. D., Geometric aspects of convex sets with the Radon-Nikodým property, Lecture Notes in Mathematics, vol. 993, Springer-Verlag, Berlin, 1983. https://doi.org/10.1007/BFb0069321
Gurariy, V. I. and Lusky, W., Geometry of Müntz spaces and related questions, Lecture Notes in Mathematics, vol. 1870, Springer-Verlag, Berlin, 2005. https://doi.org/10.1007/11551621
Haller, R., Langemets, J., Lima, V., and Nadel, R., Symmetric strong diameter two property, Mediterr. J. Math. 16 (2019), no. 2, paper no. 35, 17 pp. https://doi.org/10.1007/s00009-019-1306-1
Haller, R., Langemets, J., and Nadel, R., Stability of average roughness, octahedrality, and strong diameter $2$ properties of Banach spaces with respect to absolute sums, Banach J. Math. Anal. 12 (2018), no. 1, 222–239. https://doi.org/10.1215/17358787-2017-0040
Haller, R., Langemets, J., and Põldvere, M., On duality of diameter $2$ properties, J. Convex Anal. 22 (2015), no. 2, 465–483.
Klee, V., Polyhedral sections of convex bodies, Acta Math. 103 (1960), 243–267. https://doi.org/10.1007/BF02546358
Werner, D., A remark about Müntz spaces, http://page.mi.fu-berlin.de/werner99/preprints/muentz.pdf, 2008.
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Published
2020-09-03
How to Cite
Martiny, A. (2020). On octahedrality and Müntz spaces. MATHEMATICA SCANDINAVICA, 126(3), 513–518. https://doi.org/10.7146/math.scand.a-119844
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