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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
MathSciNet review: 0341032
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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],

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

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  • [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  • [6] A. Pełczyński, Collected letters (unpublished).

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Keywords: Approximation property, nuclear operators
Article copyright: © Copyright 1973 American Mathematical Society

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