Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On projections in power series spaces and the existence of bases


Author: Jörg Krone
Journal: Proc. Amer. Math. Soc. 105 (1989), 350-355
MSC: Primary 46A45; Secondary 46A35
DOI: https://doi.org/10.1090/S0002-9939-1989-0933516-5
MathSciNet review: 933516
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Mityagin posed the problem, whether complemented subspaces of nuclear infinite type power series spaces have a basis. A related more general question was asked by Pełczyński. It is well known for a complemented subspace $ E$ of a nuclear infinite type power series space, that its diametral dimension can be represented by $ \Delta E = \Delta {\Lambda _\infty }(\alpha )$ for a suitable sequence $ \alpha $ with $ {\alpha _j} \geq \ln (j + 1)$. In this article we prove the existence of a basis for $ E$ in case that $ {\alpha _j} \geq j$ and $ \sup \tfrac{{{\alpha _{2j}}}}{{{\alpha _j}}} < \infty $.


References [Enhancements On Off] (What's this?)

  • [1] Ed Dubinsky, The structure of nuclear Fréchet spaces, Lecture Notes in Mathematics, vol. 720, Springer, Berlin, 1979. MR 537039
  • [2] Ed Dubinsky and Dietmar Vogt, Bases in complemented subspaces of power series spaces, Bull. Polish Acad. Sci. Math. 34 (1986), no. 1-2, 65–67 (English, with Russian summary). MR 850315
  • [3] B. S. Mityagin, Sur l'equivalence des bases inconditionelles dans les echelles de Hilbert, C. R. Acad. Sci. Paris 269 (1969), 426-428.
  • [4] B. S. Mitjagin, Equivalence of bases in Hilbert scales, Studia Math. 37 (1970/71), 111–137 (Russian). MR 0322470
  • [5] B. S. Mitjagin and G. M. Henkin, Linear problems of complex analysis, Uspehi Mat. Nauk 26 (1971), no. 4 (160), 93–152 (Russian). MR 0287297
  • [6] A. Pełczyński, Problem 37, Studia Math. 38 (1970), 476.
  • [7] Tosun Terzioğlu, Die diametrale Dimension von lokalkonvexen Räumen, Collect. Math. 20 (1969), 49–99 (German). MR 0253016
  • [8] T. Terzioğlu, On the diametral dimension of some classes of 𝐹-spaces, J. Karadeniz Univ. Fac. Arts Sci. Ser. Math.-Phys. 8 (1985), 1–13 (English, with Turkish summary). MR 924467
  • [9] Dietmar Vogt, Charakterisierung der Unterräume von 𝑠, Math. Z. 155 (1977), no. 2, 109–117. MR 0463885, https://doi.org/10.1007/BF01214210
  • [10] Dietmar Vogt, Ein Isomorphiesatz für Potenzreihenräume, Arch. Math. (Basel) 38 (1982), no. 6, 540–548 (German). MR 668058, https://doi.org/10.1007/BF01304829
  • [11] Dietmar Vogt, Kernels of Eidelheit matrices and related topics, Proceedings of the functional analysis conference (Silivri/Istanbul, 1985), 1986, pp. 232–256 (English, with Turkish summary). MR 872881
  • [12] Dietmar Vogt and Max Josef Wagner, Charakterisierung der Quotientenräume von 𝑠 und eine Vermutung von Martineau, Studia Math. 67 (1980), no. 3, 225–240 (German, with English summary). MR 592388
  • [13] Dietmar Vogt and Max-Josef Wagner, Charakterisierung der Unterräume und Quotientenräume der nuklearen stabilen Potenzreihenräume von unendlichem Typ, Studia Math. 70 (1981), no. 1, 63–80 (German, with English summary). MR 646960
  • [14] M. J. Wagner, Stable complemented subspaces of $ (s)$ have a basis, Seminar Lecture, AG Funktionalanalysis Düsseldorf/Wuppertal, 1985.
  • [15] V. D. Zaharjuta, Isomorphism of spaces of analytic functions, Soviet Math. Dokl. 22 (1980), 631-634.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46A45, 46A35

Retrieve articles in all journals with MSC: 46A45, 46A35


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0933516-5
Article copyright: © Copyright 1989 American Mathematical Society