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Remarks on symmetric smooth norms

  • P. H´Jek (a1) and V. Zizler (a1)

Abstract

A characterisation is given of separable Banach spaces with symmetric Schauder bases which admit equivalent symmetric norms that are Gâteaux differentiable or uniformly rotund in every direction. Some applications to questions on distortion of norms on l are discussed.

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References

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[1]Argyros, S. and Farmaki, V., ‘On the structure of weakly compact subsets of Hilbert spaces and applications to the geometry of Banach spaces’, Trans. Amer. Math. Soc. 289 (1985), 409427.
[2]Deville, R., Godefroy, G. and Zizler, V., Smoothness and renormings in Banach spaces, Monographs Surveys Pure Appl. Maths. 64 (Pitman, London, 1993).
[3]Hájek, P., ‘Polynomials and injections of Banach spaces into superreflexive spaces’, Arch. Math. 63 (1994), 3944.
[4]Hájek, P., ‘On convex functions in co(w1)1, Collect. Math. (to appear).
[5]Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces I. Sequence spaces (Springer-Verlag, Berlin, Heidelberg, New York, 1977).
[6]Troyanski, S., ‘On nonseparable Banach space with a symmetric basisStudio. Math. 53 (1975), 253263.
[7]Troyanski, S., ‘Gâteaux differentiable norms in LpMath. Ann. 287 (1990), 221227.
[8]Troyanski, S., ‘Equivalent norms in nonseparable Banach spaces with unconditional basis’, Teor. Funktsii˘ Funktsional Anal, i Prilozhen 6 (1968), 5965.
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Remarks on symmetric smooth norms

  • P. H´Jek (a1) and V. Zizler (a1)

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