RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
 General information Latest issue Archive Impact factor Subscription License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Uspekhi Mat. Nauk: Year: Volume: Issue: Page: Find

 Uspekhi Mat. Nauk, 1970, Volume 25, Issue 3(153), Pages 113–174 (Mi umn5344)

Geometric theory of Banach spaces. Part I. The theory of basis and minimal systems

V. D. Milman

Abstract: In recent years substantial success has been achieved in the study of geometric and linear topological properties of Banach spaces ($B$-spaces). Our aim is to present some of the results that have been obtained since the appearance of the well-known survey of the geometric theory of $B$-spaces, the monograph of M. M. Day “Normed Linear Spaces”.
The term “geometric theory” which we use is to a large extent conventional. At present the principal method of investigating $B$-spaces is to study special sequences of elements of a space; this is more reminiscent of the methods of analysis than of geometry. In the first part of the survey we give an account of the apparatus of the theory of sequences and demonstrate its potential in investigating topological properties of Banach spaces. At the same time a general look at the whole host of facts, which make the current theory so rich, becomes possible in the study of the geometric structure of the unit sphere, that is, of the isometric properties of a space. This approach to the investigation of Banach spaces will be developed in a second part. These two parts do not exhaust the contemporary theory of normed linear spaces, which consists of at least two other large branches: the finite-dimensional Banach spaces or Minkowski spaces and the investigation of isomorphisms and embeddings. Each of these domains has recently received a fundamental stimulus to its development.
It is sufficient for the reader of the present article to be acquainted with the elements of functional analysis as given in Chapters 1–5 of [72] or in Chapters 1–4 of [45]. We shall omit the proofs of statements that are given in sufficient detail in the Russian literature, or that can be obtained by methods illustrated by other examples. In addition, we shall not mention proofs that would lead us away from the exposition of the method.

Full text: PDF file (7062 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 1970, 25:3, 111–170

Bibliographic databases:

UDC: 519.9
MSC: 46A03, 46B04, 46A45, 40A30

Citation: V. D. Milman, “Geometric theory of Banach spaces. Part I. The theory of basis and minimal systems”, Uspekhi Mat. Nauk, 25:3(153) (1970), 113–174; Russian Math. Surveys, 25:3 (1970), 111–170

Citation in format AMSBIB
\Bibitem{Mil70} \by V.~D.~Milman \paper Geometric theory of Banach spaces. Part~I. The theory of basis and minimal systems \jour Uspekhi Mat. Nauk \yr 1970 \vol 25 \issue 3(153) \pages 113--174 \mathnet{http://mi.mathnet.ru/umn5344} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=280985} \zmath{https://zbmath.org/?q=an:0198.16503|0221.46015} \transl \jour Russian Math. Surveys \yr 1970 \vol 25 \issue 3 \pages 111--170 \crossref{https://doi.org/10.1070/RM1970v025n03ABEH003790} 

• http://mi.mathnet.ru/eng/umn5344
• http://mi.mathnet.ru/eng/umn/v25/i3/p113

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles
Erratum
Cycle of papers

This publication is cited in the following articles:
1. B. S. Mityagin, “The homotopy structure of the linear group of a Banach space”, Russian Math. Surveys, 25:5 (1970), 59–103
2. J. Lindenstrauss, L. Tzafriri, “On orlicz sequence spaces”, Isr J Math, 10:3 (1971), 379
3. V. D. Milman, “Geometric theory of Banach spaces. Part II. Geometry of the unit sphere”, Russian Math. Surveys, 26:6 (1971), 79–163
4. A. V. Babin, “Finite dimensionality of the kernel and cokernel of quasilinear elliptic mappings”, Math. USSR-Sb., 22:3 (1974), 427–455
5. V. S. Klimov, “On functionals with an infinite number of critical values”, Math. USSR-Sb., 29:1 (1976), 91–104
6. S. A. Rakov, “Ultraproducts and the “three spaces problem””, Funct. Anal. Appl., 11:3 (1977), 236–237
7. S. Guerre, J. T. Lapresté, “Quelques proprietes des espaces de Banach stables”, Isr J Math, 39:3 (1981), 247
8. A. N. Plichko, “Selection of subspaces with special properties in a Banach space and some properties of quasicomplements”, Funct. Anal. Appl., 15:1 (1981), 67–68
9. Paolo Terenzi, “Sistemi biortogonali negli spazi di Banach”, Seminario Mat e Fis di Milano, 53:1 (1983), 83
10. E. N. Domanskii, “On the equivalence of convergence of a regularizing algorithm to the existence of a solution to an ill-posed problem”, Russian Math. Surveys, 42:5 (1987), 123–144
11. A. M. Sedletskii, “Projection from the spaces $E^p$ on a convex polygon onto subspaces of periodic functions”, Math. USSR-Izv., 33:2 (1989), 373–390
12. Eve Oja, Dirk Werner, “Remarks onM-Ideals of Compact Operators onX ⊕p X”, Math Nachr, 152:1 (1991), 101
13. F. García-Castellón, “Relative Position, Sequences and Operators in Banach Spaces”, Math Nachr, 166:1 (1994), 135
14. Antonio Plans, “Optimization of sequences in a Banach space”, Seminario Mat e Fis di Milano, 64:1 (1994), 45
15. V. S. Balaganskii, “Smooth antiproximinal sets”, Math. Notes, 63:3 (1998), 415–418
16. R. V. Vershinin, “On $(1+\varepsilon_n)$-bounded $M$-bases”, Russian Math. (Iz. VUZ), 43:4 (1999), 22–25
17. A. M. Sedletskii, “Analytic Fourier Transforms and Exponential Approximations. I”, Journal of Mathematical Sciences, 129:6 (2005), 4251–4408
18. T.Domı́nguez Benavides, M.A.Japón Pineda, S. Prus, “Weak compactness and fixed point property for affine mappings”, Journal of Functional Analysis, 209:1 (2004), 1
19. V. I. Filippov, “Perturbation of the trigonometric system in $L_1(0,\pi)$”, Math. Notes, 80:3 (2006), 410–416
20. A. M. Sedletskii, “On the Stability of the Uniform Minimality of a Set of Exponentials”, Journal of Mathematical Sciences, 155:1 (2008), 170–182
21. Ioannis A. Polyrakis, “Demand functions and reflexivity”, Journal of Mathematical Analysis and Applications, 338:1 (2008), 695
22. Antonio Plans, Dolores Lerís, “On the Action of a Linear Operator over Sequences in a Banach Space”, Math Nachr, 180:1 (2009), 285
23. Ioannis A. Polyrakis, Foivos Xanthos, “Cone characterization of Grothendieck spaces and Banach spaces containing c 0”, Positivity, 2010
24. E. Casini, E. Miglierina, “Cones with bounded and unbounded bases and reflexivity”, Nonlinear Analysis: Theory, Methods & Applications, 72:5 (2010), 2356
25. Elói Medina Galego, “The C(K,X) spaces for compact metric spaces K and X with a uniformly convex maximal factor”, Journal of Mathematical Analysis and Applications, 2011
26. A. Sh. Shukurov, “About one type of sequences that are not a Schauder basis in Hilbert spaces”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 25:2 (2015), 244–247
27. Ufa Math. J., 9:1 (2017), 109–122
•  Number of views: This page: 900 Full text: 436 References: 46 First page: 1