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
J. Hagler and W. B. Johnson, On Banach spaces whose dual balls are not weak* sequentially compact, Israel J. Math. 28(1977), 325–330.
R. H. Lohman, Gaps between spheres in normed linear spaces, Canad. Math. Bull. 23(1980), 347–354.
A. Nissenzweig, w*sequential convergence, Israel J. Math. 22(1975), 266–272.
H. P. Rosenthal, A characterization of Banach spaces containing l1, Proc. Nat. Acad. Sci. U. S. A. 71(1974), 2411–2413.
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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
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