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Abstract

In the last years the theory of bases in Banach spaces has rapidly developped*). We shall make here some remarks on a few recent results and raise some new problems in this field. The paper has partially an expository character, since we want to make it accessible also to those who are not assumed to know basis theory.

Keywords

Banach Space Studia Math Separable Banach Space Unconditional Basis Topological Linear Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    E. J. Akutowicz, Construction of a Schauder basis in some spaces of holomorphic functions in the unit disc. Colloq. Math. 15 (1966), 287–296.Google Scholar
  2. [2]
    K. I. Babenko, On conjugate functions. Dokl. Akad. Nauk. SSSR 57 (1948), 157–160 (Russian).Google Scholar
  3. [3]
    S. Banach, Théorie des opérations linéaires. Monografie Matematyczne, Warszawa 1932.Google Scholar
  4. [4]
    S. Banach und S. Mazur, Zur Theorie der linearen Dimension. Studia Math. 4 (1933), 100–112.Google Scholar
  5. [5]
    N. K. Bari, Biorthogonal systems and bases in Hilbert space. Moskov. Gos. Univ. Uceneye Zapiski 148, Matematika 4 (1951), 69–107 (Russian).Google Scholar
  6. [6]
    C. Bessaga and A. Pelczynski, On bases and unconditional convergence of series in Banach spaces. Studia Math. 17 (1958)151–164.Google Scholar
  7. [7]
    H. F. Bohnenblust, Subspaces of l p,n spaces. Amer. J. Math. 63 (1941)64–72.CrossRefGoogle Scholar
  8. [8]
    A. Grothendieck, Sur les applications linéaires faiblement compactes d’espaces du type C(K). Canad. J. Math. 5 (1953)129–173.CrossRefGoogle Scholar
  9. [9]
    A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires. Mem. Amer. Math. Soc. 16 (1955)140 pp.Google Scholar
  10. [10]
    I. M. Gelfand, Remark on the paper of N. K. Bari “Biorthogonal systems and bases in Hilbert space”. Moscov. Gos. Univ. Uceneye Zapiski 148, Matematika 4, (1951), 224–225 (Russian)Google Scholar
  11. [11]
    V. I. Gurarii, On some geometric characteristics of subspaces and of bases in Banach spaces Coll. Math. 13 (1964)59–63 (Russian).Google Scholar
  12. [l2]
    I. Kaplansky, Functional Analysis. In: Surveys in Applied Math. IV, 3–34. Wiley, New York; Chapman and Hall, London 1957.Google Scholar
  13. [13]
    G. Köthe, Probleme der linearen Algebra in topologischen Vektorräumen. Proc. Internat. Sympos. on Linear Spaces (July 5–12, 1960), 290–298. Jerusalem Academic Press; Pergamon, Oxford 1961.Google Scholar
  14. [14]
    J. Lindenstrauss and A. Pelczynski, Absolutely summing operators in 𝓛-spaces and their applications. Studia Math. 39 (1968)275–326.Google Scholar
  15. [15]
    J. Lindenstrauss and M. Zippin, Banach spaces with a unique unconditional basis. (To appear.)Google Scholar
  16. [16]
    A. Pelczynski, Projections in certain Banach spaces. Studia Math. 19 (1960)209–228.Google Scholar
  17. [17]
    A. Pelczynski, Some open questions in functional analysis. A lecture given to Louisiana State University (dittoed notes), 1966.Google Scholar
  18. [18]
    A. Pelczynski, Universal bases. Studia Math. (To appear.)Google Scholar
  19. [19]
    A. Pelczynski and I. Singer, On non-equivalent bases and conditional bases in Banach spaces. Studia Math. 25 (1964)5–25.Google Scholar
  20. [20]
    J. Schauder, Zur Theorie des stetigen Abbildungen in Funktionalräumen. Math. Z. 26 (1927) 47–65.CrossRefGoogle Scholar
  21. [21]
    I. Singer, On Banach spaces with a symmetric basis. Rev. Math. Pures Appl. 6 (1961)159–166 (Russian).Google Scholar
  22. [22]
    I. Singer, Some characterizations of symmetric bases in Banach spaces. Bull. Acad. Polon. Sci.Sér. Sci. Math. Astronom. Phys. 10 (1962)185–192.Google Scholar
  23. [23]
    I. Singer, Bases in Banach spaces. I, Studii si cercet. mat. 14 (1963)533–585Google Scholar
  24. [23a]
    I. Singer, Bases in Banach spaces. II, Studii si cercet. mat., 15 (1964)157–208Google Scholar
  25. [23b]
    I. Singer, Bases in Banach spaces. III, Studii si cercet. mat., 15 (1964)675–725 (Romanian).Google Scholar
  26. [24]
    I. Singer, On the basis problem in topological linear spaces. Rev. Roumaine Math. Pures Appl. 10 (1965)453–457.Google Scholar
  27. [25]
    I. Singer, On a theorem of N. K. Bari and I. M. Gelfand. Arch. Math. (To appear.)Google Scholar
  28. [26]
    I. Singer, Bases in Banach spaces (monograph). To appear.Google Scholar
  29. [27]
    M. Zippin, On a certain basis in c o. Israel J. Math. 4 (1966)199–204.CrossRefGoogle Scholar
  30. [28]
    M. Zippin, On perfectly homogeneous bases in Banach spaces. Israel J. Math. 4 (1966)265–272.CrossRefGoogle Scholar
  31. [29]
    M. Zippin, A remark on bases and reflexivity in Banach spaces. Israel J. Math. 6 (1968), 74–79.CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 1969

Authors and Affiliations

  • Ivan Singer
    • 1
  1. 1.Institutul de MatematicaBucarestRomania

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