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

This article is cited in 27 scientific papers (total in 27 papers)

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.

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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}


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    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  mathnet  crossref  mathscinet  zmath
    2. J. Lindenstrauss, L. Tzafriri, “On orlicz sequence spaces”, Isr J Math, 10:3 (1971), 379  crossref  mathscinet  zmath
    3. V. D. Milman, “Geometric theory of Banach spaces. Part II. Geometry of the unit sphere”, Russian Math. Surveys, 26:6 (1971), 79–163  mathnet  crossref  mathscinet  zmath
    4. A. V. Babin, “Finite dimensionality of the kernel and cokernel of quasilinear elliptic mappings”, Math. USSR-Sb., 22:3 (1974), 427–455  mathnet  crossref  mathscinet  zmath
    5. V. S. Klimov, “On functionals with an infinite number of critical values”, Math. USSR-Sb., 29:1 (1976), 91–104  mathnet  crossref  mathscinet  zmath  isi
    6. S. A. Rakov, “Ultraproducts and the “three spaces problem””, Funct. Anal. Appl., 11:3 (1977), 236–237  mathnet  crossref  mathscinet  zmath
    7. S. Guerre, J. T. Lapresté, “Quelques proprietes des espaces de Banach stables”, Isr J Math, 39:3 (1981), 247  crossref  mathscinet  zmath  isi
    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  mathnet  crossref  mathscinet  zmath  isi
    9. Paolo Terenzi, “Sistemi biortogonali negli spazi di Banach”, Seminario Mat e Fis di Milano, 53:1 (1983), 83  crossref  mathscinet  zmath
    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  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    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  mathnet  crossref  mathscinet  zmath
    12. Eve Oja, Dirk Werner, “Remarks onM-Ideals of Compact Operators onX ⊕p X”, Math Nachr, 152:1 (1991), 101  crossref  mathscinet  zmath  isi
    13. F. García-Castellón, “Relative Position, Sequences and Operators in Banach Spaces”, Math Nachr, 166:1 (1994), 135  crossref  mathscinet  zmath  isi
    14. Antonio Plans, “Optimization of sequences in a Banach space”, Seminario Mat e Fis di Milano, 64:1 (1994), 45  crossref  mathscinet
    15. V. S. Balaganskii, “Smooth antiproximinal sets”, Math. Notes, 63:3 (1998), 415–418  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    16. R. V. Vershinin, “On $(1+\varepsilon_n)$-bounded $M$-bases”, Russian Math. (Iz. VUZ), 43:4 (1999), 22–25  mathnet  mathscinet  zmath
    17. A. M. Sedletskii, “Analytic Fourier Transforms and Exponential Approximations. I”, Journal of Mathematical Sciences, 129:6 (2005), 4251–4408  mathnet  crossref  mathscinet  zmath
    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  crossref
    19. V. I. Filippov, “Perturbation of the trigonometric system in $L_1(0,\pi)$”, Math. Notes, 80:3 (2006), 410–416  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    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  mathnet  crossref  mathscinet  zmath  elib
    21. Ioannis A. Polyrakis, “Demand functions and reflexivity”, Journal of Mathematical Analysis and Applications, 338:1 (2008), 695  crossref
    22. Antonio Plans, Dolores Lerís, “On the Action of a Linear Operator over Sequences in a Banach Space”, Math Nachr, 180:1 (2009), 285  crossref
    23. Ioannis A. Polyrakis, Foivos Xanthos, “Cone characterization of Grothendieck spaces and Banach spaces containing c 0”, Positivity, 2010  crossref
    24. E. Casini, E. Miglierina, “Cones with bounded and unbounded bases and reflexivity”, Nonlinear Analysis: Theory, Methods & Applications, 72:5 (2010), 2356  crossref
    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  crossref
    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  mathnet  elib
    27. Ufa Math. J., 9:1 (2017), 109–122  mathnet  crossref  isi  elib
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