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Nonexpansive projections onto two-dimensional subspaces of Banach spaces

  • Bruce Calvert (a1) and Simon Fitzpatrick (a1)

Abstract

We show that if a three dimensional normed space X has two linearly independent smooth points e and f such that every two-dimensional subspace containing e or f is the range of a nonexpansive projection then X is isometrically isomorphic to ℓp(3) for some p, 1 < p ≤ ∞. This leads to a characterisation of the Banach spaces c0 and ℓp, 1 < p ≤ ∞, and a characterisation of real Hilbert spaces.

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Copyright

References

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[1]Ando, T., ‘Banachverbande und positive projection’, Math. Z. 109 (1969), 121130.
[2]Blaschke, W., ‘Raumliche Variationsprobleme mit symmetrischer Transversalitatsbedingung’, Leipziger Berichte, Math-phys. Klasse 68 (1916), 5055.
[3]Calvert, Bruce and Fitzpatrick, Simon, ‘A Banach lattice characterisation of c 0 and ℓp’, (submitted).
[4]Calvert, Bruce and Fitzpatrick, Simon, ‘Consequences of time duality map taking planes to planes’ (to appear), in Proceedings of the VII International Conference on Nonlinear Analysis and Applications, ed. Lakshmikantham, V. to appear.
[5]Calvert, Bruce and Fitzpatrick, Simon, ‘Characterising ℓp and c 0 by projections onto hyperplanes’, Boll. Un. Math. Ital. (to appear).
[6]Lacy, H.E., The Isometric Theory of Classical Banach Spaces (Springer-Verlag, Berlin, 1974).
[7]Lindenstrauss, J. and Tzafriri, L., Classical Banach Spaces I (Springer-Verlag, Berlin, 1977).
[8]Lindenstrauss, J. and Tzafriri, L., Classical Banach Spaces II (Springer-Verlag, Berlin, 1979).
[9]Roberts, A. Wayne and Varberg, Dale E., Convez Functions (Academic Press, New York, 1973).
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