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)

 
 

 

Unconditional bases in countable- $ \mathcal{L}_1$ spaces


Author: J. C. Díaz
Journal: Proc. Amer. Math. Soc. 106 (1989), 357-363
MSC: Primary 46A35; Secondary 46A12
DOI: https://doi.org/10.1090/S0002-9939-1989-0931728-8
MathSciNet review: 931728
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we show that a countably- $ {\mathcal{L}_1}$ space which has an unconditional basis is isomorphic to some echelon sequence space of order 1. As a consequence, a countably- $ {\mathcal{L}_1}$ space with a basis is nuclear if all its bases are unconditional (this gives a partial answer to a conjecture of Wojtynski). We also study those countably- $ {\mathcal{L}_1}$ spaces on which a Fréchet lattice structure can be defined.


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

  • [1] H. Jarchow, Locally convex spaces, Teubner, Stuttgart, 1981. MR 632257 (83h:46008)
  • [2] Y. Komura and S. Koshi, Nuclear vector lattices, Math. Ann. 163 (1966), 105-110. MR 0192327 (33:553)
  • [3] J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in $ {\mathcal{L}_p}$-spaces and their applications, Studia Math. 29 (1968), 275-326. MR 0231188 (37:6743)
  • [4] J. A. Lopez-Molina, The dual and bidual of an echelon Köthe space, Collect. Math. 31 (1980), 159-191. MR 611558 (83f:46028)
  • [5] -, Reflexividad en los espacios escalonados de Köthe, Rev. Real Acad. Cienc. Exact Madrid 75 (1981), 213-232.
  • [6] H. H. Schaefer, Topological vector spaces, Springer-Verlag, 1971. MR 0342978 (49:7722)
  • [7] -, Banach lattices and positive operators, Springer-Verlag, 1974. MR 0423039 (54:11023)
  • [8] L. J. Weill, Unconditional and shrinking bases in locally convex spaces, Pacific J. Math. 29 (1969), 467-483. MR 0246083 (39:7389)
  • [9] W. Wojtynski, On conditional bases in non-nuclear Fréchet spaces, Studia Math. 35 (1970), 77-96. MR 0271695 (42:6576)
  • [10] A. C. Zaanen, Integration, North-Holland, 1967. MR 0222234 (36:5286)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46A35, 46A12

Retrieve articles in all journals with MSC: 46A35, 46A12


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0931728-8
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society