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Orthogonality, Projections and Equivalent Bases

  • Jürg T. Marti
Part of the Springer Tracts in Natural Philosophy book series (STPHI, volume 18)

Abstract

If there is a basis for a closed linear subspace Y of Banach space X, a very general condition allows to define a projection of X on Y On the other hand, projections are very useful tools for existence proofs of bases. Indeed, they can be applied in the proof of the existence theorem of Nikol’Skiĭ given in the first paragraph. The theorem is quite essential in the theory and in the application of the theory of bases and, accordingly, we have to make reference to it in many of the subsequent theorems and corollaries. The second paragraph refers to the very nice fact that the concept of orthogonality can be extended from Hilbert spaces to the normed linear spaces. It gives the relation between total orthogonal systems, simple N 1-spaces and monotone bases. The last section is concerned with equivalent bases, block bases and the theorem that every infinite dimensional Banach space contains an infinite dimensional subspace with a basis.

Keywords

Banach Space Block Basis Normed Linear Space Separable Banach Space Equivalent Basis 
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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Reference

  1. Bessaga, C. Bases in certain spaces of continuous functions. Bull. Acad. Pol. Sci. CHI, 5, 11–14(1957).MathSciNetMATHGoogle Scholar
  2. Bessaga, C., and A. Pelczynski On bases and unconditional convergence of series in Banach spaces. Studia Math. 17, 151–164(1958).MathSciNetMATHGoogle Scholar
  3. Day, M. M. On the basis problem in normed spaces. Proc. Amer. Math. Soc. 13, 655–658 (1962).MathSciNetMATHCrossRefGoogle Scholar
  4. Gelbaum, B. R. Expansions in Banach spaces. Duke Math. J. 17, 187–196 (1950).MathSciNetMATHCrossRefGoogle Scholar
  5. Grinblyum, M. M. Certains théorèmes sur la base dans un espace du type (B). Doklady Akad. Nauk SSSR (N. S.) 31, 428–432 (1941) (Russian).Google Scholar
  6. Grinblyum, M. M. Certains théorèmes sur la base dans un espace du type (B). Math. Rev. 3, 49 (1942).Google Scholar
  7. James, R. C. Orthogonality in normed linear spaces. Duke Math. J. 12, 291–302 (1945).MathSciNetMATHCrossRefGoogle Scholar
  8. James, R. C. Orthogonality and linear functionals in normed linear spaces. Trans. Amer. Math. Soc. 61, 265–292 (1947).MathSciNetCrossRefGoogle Scholar
  9. James, R. C. Inner products in normed linear spaces. Bull. Amer. Math. Soc. 53, 559–566 (1947).MathSciNetMATHCrossRefGoogle Scholar
  10. Lindenstrauss, J. Extension of compact operators. Mem. Amer. Math. Soc. 48 (1964).Google Scholar
  11. Michael, E., and A. Pelczynski Separable Banach spaces which admit ln approximations. Israel Math. J. 4, 189–198(1966).MathSciNetMATHCrossRefGoogle Scholar
  12. Nikol’skiǐ, V. N. The best approximation and a basis in a Fréchet space. Doklady Akad. Nauk SSSR (N. S.) 59, 639–642 (1948) (Russian).MathSciNetGoogle Scholar
  13. Foias, C., and I. Singer Some remarks on strongly linearly independent sequences and bases in Banach spaces. Rev. math. pures appl. 6, 589–594 (1961).MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 1969

Authors and Affiliations

  • Jürg T. Marti
    • 1
  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA

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