Book Review
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MathSciNet review:
3686329
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Book Information:
Authors:
Shiri Artstein-Avidan,
Apostolos Giannopoulos and
Vitali D. Milman
Title:
Asymptotic geometric analysis, Part I
Additional book information:
Mathematical Surveys and Monographs, Vol. 202,
American Mathematical Society,
Providence, RI,
2015,
xx+451 pp.,
ISBN 978-1-4704-2193-9
Stefan Banach, Théorie des opérations linéaires, Chelsea Publishing Co., New York, 1955 (French). MR 0071726
Silouanos Brazitikos, Apostolos Giannopoulos, Petros Valettas, and Beatrice-Helen Vritsiou, Geometry of isotropic convex bodies, Mathematical Surveys and Monographs, vol. 196, American Mathematical Society, Providence, RI, 2014. MR 3185453, DOI 10.1090/surv/196
Aryeh Dvoretzky, Some results on convex bodies and Banach spaces, Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960) Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, 1961, pp. 123–160. MR 0139079
A. Dvoretzky and C. A. Rogers, Absolute and unconditional convergence in normed linear spaces, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 192–197. MR 33975, DOI 10.1073/pnas.36.3.192
T. Figiel, A short proof of Dvoretzky’s theorem, Séminaire Maurey-Schwartz 1974–1975: Espaces $L^{p}$, applications radonifiantes et géométrie des espaces de Banach, Exp. No. XXIII, Centre Math., École Polytech., Paris, 1975, pp. 6 pp. (erratum, p. 3). MR 0407566
A. Grothendieck, Résumé des résultats essentiels dans la théorie des produits tensoriels topologiques et des espaces nucléaires, Ann. Inst. Fourier (Grenoble) 4 (1952), 73–112 (1954) (French). MR 61754
A. Grothendieck, Sur certaines classes de suites dans les espaces de Banach et le théorème de Dvoretzky-Rogers, Bol. Soc. Mat. São Paulo 8 (1953), 81–110 (1956) (French). MR 94683
Paul Lévy, Problèmes concrets d’analyse fonctionnelle. Avec un complément sur les fonctionnelles analytiques par F. Pellegrino, Gauthier-Villars, Paris, 1951 (French). 2d ed. MR 0041346
J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in $L_{p}$-spaces and their applications, Studia Math. 29 (1968), 275–326. MR 231188, DOI 10.4064/sm-29-3-275-326
J. Lindenstrauss and H. P. Rosenthal, The ${\cal L}_{p}$ spaces, Israel J. Math. 7 (1969), 325–349. MR 270119, DOI 10.1007/BF02788865
V. D. Milman, A new proof of A. Dvoretzky’s theorem on cross-sections of convex bodies, Funkcional. Anal. i Priložen. 5 (1971), no. 4, 28–37 (Russian). MR 0293374
Vitali D. Milman and Gideon Schechtman, Asymptotic theory of finite-dimensional normed spaces, Lecture Notes in Mathematics, vol. 1200, Springer-Verlag, Berlin, 1986. With an appendix by M. Gromov. MR 856576
Gilles Pisier, The volume of convex bodies and Banach space geometry, Cambridge Tracts in Mathematics, vol. 94, Cambridge University Press, Cambridge, 1989. MR 1036275, DOI 10.1017/CBO9780511662454
Andrzej Szankowski, On Dvoretzky’s theorem on almost spherical sections of convex bodies, Israel J. Math. 17 (1974), 325–338. MR 350388, DOI 10.1007/BF02756881
L. Tzafriri, On Banach spaces with unconditional bases, Israel J. Math. 17 (1974), 84–93. MR 348456, DOI 10.1007/BF02756829
References
- S. Banach, Théorie des opérations linéaires, Chelsea Publishing Co., New York, 1955 (French). MR 0071726
- S. Brazitikos, A. Giannopoulos, P. Valettas, and B.-H. Vritsiou, Geometry of isotropic convex bodies, Mathematical Surveys and Monographs, vol. 196, American Mathematical Society, Providence, RI, 2014. MR 3185453, DOI 10.1090/surv/196
- A. Dvoretzky, Some results on convex bodies and Banach spaces, Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960) Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, 1961, pp. 123–160. MR 0139079
- A. Dvoretzky and C. A. Rogers, Absolute and unconditional convergence in normed linear spaces, Proc. Nat. Acad. Sci. U. S. A. 36 (1950), 192–197. MR 0033975
- T. Figiel, A short proof of Dvoretzky’s theorem, Séminaire Maurey-Schwartz 1974–1975: Espaces $L^{p}$, applications radonifiantes et géométrie des espaces de Banach, Exp. No. XXIII, Centre Math., École Polytech., Paris, 1975, pp. 6 pp. (erratum, p. 3). MR 0407566
- A. Grothendieck, Résumé des résultats essentiels dans la théorie des produits tensoriels topologiques et des espaces nucléaires, Ann. Inst. Fourier Grenoble 4 (1952), 73–112 (1954) (French). MR 0061754
- A. Grothendieck, Sur certaines classes de suites dans les espaces de Banach et le théorème de Dvoretzky-Rogers, Bol. Soc. Mat. São Paulo 8 (1953), 81–110 (1956) (French). MR 0094683
- P. Lévy, Problèmes concrets d’analyse fonctionnelle. Avec un complément sur les fonctionnelles analytiques par F. Pellegrino, Gauthier-Villars, Paris, 1951 (French). 2d ed. MR 0041346
- J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in $L_{p}$-spaces and their applications, Studia Math. 29 (1968), 275–326. MR 0231188
- J. Lindenstrauss and H. P. Rosenthal, The ${\mathcal {L}}_{p}$ spaces, Israel J. Math. 7 (1969), 325–349. MR 0270119
- V. D. Milman, A new proof of A. Dvoretzky’s theorem on cross-sections of convex bodies, Funkcional. Anal. i Priložen. 5 (1971), no. 4, 28–37 (Russian). MR 0293374
- V. D. Milman and G. Schechtman, Asymptotic theory of finite-dimensional normed spaces, Lecture Notes in Mathematics, vol. 1200, Springer-Verlag, Berlin, 1986. With an appendix by M. Gromov. MR 856576
- G. Pisier, The volume of convex bodies and Banach space geometry, Cambridge Tracts in Mathematics, vol. 94, Cambridge University Press, Cambridge, 1989. MR 1036275, DOI 10.1017/CBO9780511662454
- A. Szankowski, On Dvoretzky’s theorem on almost spherical sections of convex bodies, Israel J. Math. 17 (1974), 325–338. MR 0350388
- L. Tzafriri, On Banach spaces with unconditional bases, Israel J. Math. 17 (1974), 84–93. MR 0348456
Review Information:
Reviewer:
Gideon Schechtman
Affiliation:
Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel
Email:
gideon@weizmann.ac.il
Journal:
Bull. Amer. Math. Soc.
54 (2017), 341-345
DOI:
https://doi.org/10.1090/bull/1558
Published electronically:
September 14, 2016
Review copyright:
© Copyright 2016
American Mathematical Society
