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Some Remarks and Problems on Bases in Banach Spaces

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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.

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References

  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. K. I. Babenko, On conjugate functions. Dokl. Akad. Nauk. SSSR 57 (1948), 157–160 (Russian).

    Google Scholar 

  3. S. Banach, Théorie des opérations linéaires. Monografie Matematyczne, Warszawa 1932.

    Google Scholar 

  4. S. Banach und S. Mazur, Zur Theorie der linearen Dimension. Studia Math. 4 (1933), 100–112.

    Google Scholar 

  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. C. Bessaga and A. Pelczynski, On bases and unconditional convergence of series in Banach spaces. Studia Math. 17 (1958)151–164.

    Google Scholar 

  7. H. F. Bohnenblust, Subspaces of l p,n spaces. Amer. J. Math. 63 (1941)64–72.

    Article  Google Scholar 

  8. A. Grothendieck, Sur les applications linéaires faiblement compactes d’espaces du type C(K). Canad. J. Math. 5 (1953)129–173.

    Article  Google Scholar 

  9. A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires. Mem. Amer. Math. Soc. 16 (1955)140 pp.

    Google Scholar 

  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. 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. I. Kaplansky, Functional Analysis. In: Surveys in Applied Math. IV, 3–34. Wiley, New York; Chapman and Hall, London 1957.

    Google Scholar 

  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. J. Lindenstrauss and A. Pelczynski, Absolutely summing operators in 𝓛-spaces and their applications. Studia Math. 39 (1968)275–326.

    Google Scholar 

  15. J. Lindenstrauss and M. Zippin, Banach spaces with a unique unconditional basis. (To appear.)

    Google Scholar 

  16. A. Pelczynski, Projections in certain Banach spaces. Studia Math. 19 (1960)209–228.

    Google Scholar 

  17. A. Pelczynski, Some open questions in functional analysis. A lecture given to Louisiana State University (dittoed notes), 1966.

    Google Scholar 

  18. A. Pelczynski, Universal bases. Studia Math. (To appear.)

    Google Scholar 

  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. J. Schauder, Zur Theorie des stetigen Abbildungen in Funktionalräumen. Math. Z. 26 (1927) 47–65.

    Article  Google Scholar 

  21. I. Singer, On Banach spaces with a symmetric basis. Rev. Math. Pures Appl. 6 (1961)159–166 (Russian).

    Google Scholar 

  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. I. Singer, Bases in Banach spaces. I, Studii si cercet. mat. 14 (1963)533–585

    Google Scholar 

  24. I. Singer, Bases in Banach spaces. II, Studii si cercet. mat., 15 (1964)157–208

    Google Scholar 

  25. I. Singer, Bases in Banach spaces. III, Studii si cercet. mat., 15 (1964)675–725 (Romanian).

    Google Scholar 

  26. I. Singer, On the basis problem in topological linear spaces. Rev. Roumaine Math. Pures Appl. 10 (1965)453–457.

    Google Scholar 

  27. I. Singer, On a theorem of N. K. Bari and I. M. Gelfand. Arch. Math. (To appear.)

    Google Scholar 

  28. I. Singer, Bases in Banach spaces (monograph). To appear.

    Google Scholar 

  29. M. Zippin, On a certain basis in c o . Israel J. Math. 4 (1966)199–204.

    Article  Google Scholar 

  30. M. Zippin, On perfectly homogeneous bases in Banach spaces. Israel J. Math. 4 (1966)265–272.

    Article  Google Scholar 

  31. M. Zippin, A remark on bases and reflexivity in Banach spaces. Israel J. Math. 6 (1968), 74–79.

    Article  Google Scholar 

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P. L. Butzer B. Szőkefalvi-Nagy

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© 1969 Springer Basel AG

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Singer, I. (1969). Some Remarks and Problems on Bases in Banach Spaces. In: Butzer, P.L., Szőkefalvi-Nagy, B. (eds) Abstract Spaces and Approximation / Abstrakte Räume und Approximation. ISNM International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale D’Analyse Numérique, vol 10. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5869-4_13

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  • DOI: https://doi.org/10.1007/978-3-0348-5869-4_13

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-5871-7

  • Online ISBN: 978-3-0348-5869-4

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