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Uspekhi Mat. Nauk, 1971, Volume 26, Issue 6(162), Pages 73–149 (Mi umn5278)  

This article is cited in 19 scientific papers (total in 19 papers)

Geometric theory of Banach spaces. Part II. Geometry of the unit sphere

V. D. Milman


Abstract: Interest in a geometrical approach to the study of Banach spaces is due to the following circumstance. Banach spaces have rich linear topological properties, which are extremely convenient in applications. However, the definition of a $B$-space is inseparably linked with a norm, that is, with a fixed geometrical object – the unit ball $D(B)=\{x\in B:\|x\|\leqslant 1\}$, whereas the linear topological properties depend (by definition) only on the topology of the space, that is, on a class of bounded convex bodies. Thus, we are naturally led to the question: what can be said about the linear topological properties of a space in isometric terms, that is, whilst remaining within the framework of a given norm?
The possibility of a productive investigation in this direction is essentially an infinite-dimensional situation, since in the finite-dimensional case the linear topology of a space is uniquely determined by the dimension. In view of the simplicity of the topological properties of $n$-dimensional spaces, the aim and fundamental object of investigation are geometrical (for example, the geometry of convex bodies). In the infinite-dimensional case topological questions give rise to enough concern. In this paper I follow tradition and give the main attention to results that lie in the topological channel, although it seems to me that an intrinsic study of the geometric object (an infinite-dimensional convex body) is no less interesting.

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English version:
Russian Mathematical Surveys, 1971, 26:6, 79–163

Bibliographic databases:

UDC: 519.9
MSC: 46B10, 46B04, 46B03, 46B07, 46B20
Received: 01.03.1971

Citation: V. D. Milman, “Geometric theory of Banach spaces. Part II. Geometry of the unit sphere”, Uspekhi Mat. Nauk, 26:6(162) (1971), 73–149; Russian Math. Surveys, 26:6 (1971), 79–163

Citation in format AMSBIB
\Bibitem{Mil71}
\by V.~D.~Milman
\paper Geometric theory of Banach spaces. Part~II. Geometry of the unit sphere
\jour Uspekhi Mat. Nauk
\yr 1971
\vol 26
\issue 6(162)
\pages 73--149
\mathnet{http://mi.mathnet.ru/umn5278}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=420226}
\zmath{https://zbmath.org/?q=an:0229.46017}
\transl
\jour Russian Math. Surveys
\yr 1971
\vol 26
\issue 6
\pages 79--163
\crossref{https://doi.org/10.1070/RM1971v026n06ABEH001273}


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    2. Vu Quoc Phong, “Convex sets of almost-normal structure”, Funct. Anal. Appl., 18:2 (1984), 161–162  mathnet  crossref  mathscinet  zmath  isi
    3. Thomas Schlumprecht, “An arbitrarily distortable Banach space”, Isr J Math, 76:1-2 (1991), 81  crossref  mathscinet  zmath  isi
    4. E. Odell, Th Schlumprecht, “The distortion of Hilbert space”, GAFA Geom funct anal, 3:2 (1993), 201  crossref  mathscinet  zmath
    5. M. Baronti, E. Casini, P. L. Papini, “Antipodal pairs and the geometry of Banach spaces”, Rend Circ Mat Palermo, 42:3 (1993), 369  crossref  mathscinet  zmath
    6. Edward Odell, Thomas Schlumprecht, “The distortion problem”, Acta Math, 173:2 (1994), 259  crossref  mathscinet  zmath  isi
    7. V. V. Arestov, “Approximation of unbounded operators by bounded operators and related extremal problems”, Russian Math. Surveys, 51:6 (1996), 1093–1126  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    8. Elisabetta Maluta, Stanisław Prus, “Banach Spaces Which Are Dual tok-Uniformly Convex Spaces”, Journal of Mathematical Analysis and Applications, 209:2 (1997), 479  crossref
    9. Petr Hájek, Michal Johanis, “Characterization of reflexivity by equivalent renorming”, Journal of Functional Analysis, 211:1 (2004), 163  crossref
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    12. V. P. Kondakov, A. I. Efimov, “O klassakh prostranstv Këte, v kotorykh kazhdoe dopolnyaemoe podprostranstvo imeet bazis”, Vladikavk. matem. zhurn., 10:2 (2008), 21–29  mathnet  mathscinet  elib
    13. Milman V.D., “Geometrization of probability”, In Memory of Alexander Reznikov, Progress in Mathematics, 265, 2008, 647–667  isi
    14. Lucas L.J., Owhadi H., Ortiz M., “Rigorous Verification, Validation, Uncertainty Quantification and Certification Through Concentration-of-Measure Inequalities”, Comput. Meth. Appl. Mech. Eng., 197:51-52 (2008), 4591–4609  crossref  isi
    15. Topcu U., Lucas L.J., Owhadi H., Ortiz M., “Rigorous Uncertainty Quantification Without Integral Testing”, Reliab. Eng. Syst. Saf., 96:9, SI (2011), 1085–1091  crossref  isi
    16. T. Domínguez Benavides, “Distortion and stability of the fixed point property for non-expansive mappings”, Nonlinear Analysis: Theory, Methods & Applications, 2012  crossref
    17. N. J. Kalton, “The uniform structure of Banach spaces”, Math. Ann, 354:4 (2012), 1247  crossref
    18. Liran Rotem, “A sharp Blaschke–Santaló inequality for
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  • Успехи математических наук Russian Mathematical Surveys
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