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On the existence of spheres and dual spheres without gap points

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Part of the Lecture Notes in Mathematics book series (LNM,volume 991)

Abstract

It is shown that every infinite-dimensional Banach space (resp., non-Schur space) (X,q) admits an equivalent norm p (resp., r) such that q ≦ p (resp., r ≦ q) and gap points do not exist between the q-sphere and p-sphere (resp., r-sphere) or their duals. The same conclusion is shown to hold in the case of the Schur space l1 in its usual norm.

Keywords

  • Banach Space
  • Extreme Point
  • Usual Norm
  • Real Scalar
  • Equivalent Norm

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References

  1. J. Hagler and W. B. Johnson, On Banach spaces whose dual balls are not weak* sequentially compact, Israel J. Math. 28(1977), 325–330.

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  3. A. Nissenzweig, w*sequential convergence, Israel J. Math. 22(1975), 266–272.

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© 1983 Springer-Verlag

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Lohman, R.H. (1983). On the existence of spheres and dual spheres without gap points. In: Pietsch, A., Popa, N., Singer, I. (eds) Banach Space Theory and its Applications. Lecture Notes in Mathematics, vol 991. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0061567

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  • DOI: https://doi.org/10.1007/BFb0061567

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12298-2

  • Online ISBN: 978-3-540-39877-6

  • eBook Packages: Springer Book Archive