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Quotients of Essentially Euclidean Spaces

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  09 January 2019

Tadeusz Figiel  and
William Johnson
Tadeusz Figiel
Affiliation:
Institute of Mathematics, Polish Academy of Sciences Email: [email protected]
William Johnson
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843–3368, USA Email: [email protected]
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Abstract

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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.

Keywords

essentially euclidean

MSC classification

Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 46B07: Local theory of Banach spaces 46B99: None of the above, but in this section
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 1 , March 2019 , pp. 71 - 74
Copyright
© Canadian Mathematical Society 2018 

Footnotes

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

References

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. 138–175. https://doi.org/10.1007/BFb0090053.Google Scholar
Day, M. M., On the basis problem in normed spaces . Proc. Amer. Math. Soc. 13(1962), 655–658. https://doi.org/10.1090/S0002-9939-1962-0137987-7.Google Scholar
Johnson, W. B. and Schechtman, G., Very tight embeddings of subspaces of L p , 1⩽p < 2, into ℓ p n . Geom. Funct. Anal. 13(2003), no. 4, 845–851. https://doi.org/10.1007/s00039-003-0432-9.Google Scholar
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), 97–112.Google Scholar
Litvak, A. E., Milman, V. D., and Tomczak-Jaegermann, N., Essentially-Euclidean convex bodies . Studia Math. 196(2010), no. 3, 207–221. https://doi.org/10.4064/sm196-3-1.Google Scholar
Milman, V. D., Almost Euclidean quotient spaces of subspaces of a finite-dimensional normed space . Proc. Amer. Math. Soc. 94(1985), no. 3, 445–449. https://doi.org/10.1090/S0002-9939-1985-0787891-1.Google Scholar
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.Google Scholar