On octahedrality and Müntz spaces

  • André Martiny


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.


Abrahamsen, T. A., Linear extensions, almost isometries, and diameter two, Extracta Math. 30 (2015), no. 2, 135–151.

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.
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