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Quotients of Essentially Euclidean Spaces

  • Tadeusz Figiel (a1) and William Johnson (a2)
Abstract

A precise quantitative version of the following qualitative statement is proved: If a finite-dimensional normed space contains approximately Euclidean subspaces of all proportional dimensions, then every proportional dimensional quotient space has the same property.

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Author W. J. was supported in part by NSF DMS-1565826.

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[BKT] Bourgain, J., Kalton, N. J., and Tzafriri, L., Geometry of finite-dimensional subspaces and quotients of L p . In: Geometric aspects of functional analysis (1987–88), Lecture Notes in Math., 1376, Springer, Berlin, 1989, pp. 138175. https://doi.org/10.1007/BFb0090053.
[Day] Day, M. M., On the basis problem in normed spaces . Proc. Amer. Math. Soc. 13(1962), 655658. https://doi.org/10.1090/S0002-9939-1962-0137987-7.
[JS] Johnson, W. B. and Schechtman, G., Very tight embeddings of subspaces of L p , 1⩽p < 2, into p n . Geom. Funct. Anal. 13(2003), no. 4, 845851. https://doi.org/10.1007/s00039-003-0432-9.
[KKM] Krein, M. G., Milman, D. P., and Krasnosel’ski, M. A., On the defect numbers of linear operators in Banach space and some geometric questions . (Russian) Sbornik Trudov Inst. Acad. NAUK Uk. SSR 11(1948), 97112.
[LMT-J] Litvak, A. E., Milman, V. D., and Tomczak-Jaegermann, N., Essentially-Euclidean convex bodies . Studia Math. 196(2010), no. 3, 207221. https://doi.org/10.4064/sm196-3-1.
[Mil] Milman, V. D., Almost Euclidean quotient spaces of subspaces of a finite-dimensional normed space . Proc. Amer. Math. Soc. 94(1985), no. 3, 445449. https://doi.org/10.1090/S0002-9939-1985-0787891-1.
[Pis] Pisier, G., The volume of convex bodies and Banach space geometry. Cambridge Tracts in Mathematics, 94, Cambridge University Press, Cambridge, 1989. https://doi.org/10.1017/CBO9780511662454.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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