The approximation property does not imply the bounded approximation property
Authors:
T. Figiel and W. B. Johnson
Journal:
Proc. Amer. Math. Soc. 41 (1973), 197-200
MSC:
Primary 46B05; Secondary 47B10
DOI:
https://doi.org/10.1090/S0002-9939-1973-0341032-5
MathSciNet review:
0341032
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: There is a Banach space which has the approximation property but fails the bounded approximation property. The space can be chosen to have separable conjugate, hence there is a nonnuclear operator on the space which has nuclear adjoint. This latter result solves a problem of Grothendieck [2],
- [1] P. Enflo, A counter example to the approximation problem, Acta Math. (to appear). MR 0402468 (53:6288)
- [2] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. No. 16 (1955). MR 17, 763. MR 0075539 (17:763c)
- [3] W. B. Johnson, A complementably universal conjugate Banach space and its relation to the approximation problem, Israel J. Math. 13 (1972), 301-310. MR 0326356 (48:4700)
- [4] W. B. Johnson, H. P. Rosenthal and M. Zippin, On bases, finite dimensional decompositions, and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488-506. MR 43 #6702. MR 0280983 (43:6702)
- [5] J. Lindenstrauss, On James' paper 'Separable conjugate spaces,' Israel J. Math. 9 (1971), 279-284. MR 43 #5289. MR 0279567 (43:5289)
- [6] A. Pełczyński, Collected letters (unpublished).
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46B05, 47B10
Retrieve articles in all journals with MSC: 46B05, 47B10
Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1973-0341032-5
Keywords:
Approximation property,
nuclear operators
Article copyright:
© Copyright 1973
American Mathematical Society
