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Russian Mathematical Surveys, 1970, Volume 25, Issue 3, Pages 111–170
DOI: https://doi.org/10.1070/RM1970v025n03ABEH003790
(Mi rm5344)
 

This article is cited in 42 scientific papers (total in 42 papers)

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

V. D. Milman
References:
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.
Bibliographic databases:
Document Type: Article
UDC: 519.9
Language: English
Original paper language: Russian
Citation: V. D. Milman, “Geometric theory of Banach spaces. Part I. The theory of basis and minimal systems”, 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 Russian Math. Surveys
\yr 1970
\vol 25
\issue 3
\pages 111--170
\mathnet{http://mi.mathnet.ru/eng/rm5344}
\crossref{https://doi.org/10.1070/RM1970v025n03ABEH003790}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=280985}
\zmath{https://zbmath.org/?q=an:0198.16503|0221.46015}
Linking options:
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  • https://doi.org/10.1070/RM1970v025n03ABEH003790
  • https://www.mathnet.ru/eng/rm/v25/i3/p113
    Erratum Cycle of papers
    This publication is cited in the following 42 articles:
    1. Fernando Albiac, José L. Ansorena, “The uniqueness of unconditional basis of the 2-convexified Tsirelson space, revisited”, Banach J. Math. Anal., 18:4 (2024)  crossref
    2. Elmira Mussirepova, Abdissalam A. Sarsenbi, Abdizhahan M. Sarsenbi, “Solvability of mixed problems for the wave equation with reflection of the argument”, Math Methods in App Sciences, 45:17 (2022), 11262  crossref
    3. Elmira Mussirepova, Abdissalam Sarsenbi, Abdizhahan Sarsenbi, “The inverse problem for the heat equation with reflection of the argument and with a complex coefficient”, Bound Value Probl, 2022:1 (2022)  crossref
    4. E. A. Larionov, “On stability of bases in Hilbert spaces”, Eurasian Math. J., 11:2 (2020), 65–71  mathnet  crossref
    5. Petr Hájek, Tommaso Russo, “On densely isomorphic normed spaces”, Journal of Functional Analysis, 279:7 (2020), 108667  crossref
    6. Ufa Math. J., 9:1 (2017), 109–122  mathnet  crossref  isi  elib
    7. L. V. Kritskov, A. M. Sarsenbi, “Riesz basis property of system of root functions of second-order differential operator with involution”, Diff Equat, 53:1 (2017), 33  crossref
    8. 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
    9. 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
    10. Ioannis A. Polyrakis, Foivos Xanthos, “Cone characterization of Grothendieck spaces and Banach spaces containing c 0”, Positivity, 2010  crossref
    11. E. Casini, E. Miglierina, “Cones with bounded and unbounded bases and reflexivity”, Nonlinear Analysis: Theory, Methods & Applications, 72:5 (2010), 2356  crossref
    12. Antonio Plans, Dolores Lerís, “On the Action of a Linear Operator over Sequences in a Banach Space”, Math Nachr, 180:1 (2009), 285  crossref
    13. Ioannis A. Polyrakis, “Demand functions and reflexivity”, Journal of Mathematical Analysis and Applications, 338:1 (2008), 695  crossref
    14. 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
    15. V. I. Filippov, “Perturbation of the trigonometric system in L1(0,π)”, Math. Notes, 80:3 (2006), 410–416  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    16. V. D. Milman, Mathematical Events of the Twentieth Century, 2006, 215  crossref
    17. 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
    18. A. M. Sedletskii, “Analytic Fourier Transforms and Exponential Approximations. I”, Journal of Mathematical Sciences, 129:6 (2005), 4251–4408  mathnet  crossref  mathscinet  zmath
    19. R. V. Vershinin, “On (1+εn)-bounded M-bases”, Russian Math. (Iz. VUZ), 43:4 (1999), 22–25  mathnet  mathscinet  zmath
    20. V. S. Balaganskii, “Smooth antiproximinal sets”, Math. Notes, 63:3 (1998), 415–418  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
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