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Locally uniformly rotund renorming and decompositions of Banach spaces

  • V. Zizler (a1)

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A norm |·| of a Banach space x is called locally uniformly rotund if lim|xnx| = 0 whenever xn, xX, and . It is shown that such an equivalent norm exists on every Banach space x which possesses a projectional resolution {pα} of the identify operator, for which all (pα+1pα)X admit such norms. This applies, for example, for the dual space of a space with Fréchet differentiable norm.

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References

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[1]Amir, D. and Lindenstrauss, J., “The structure of weakly compact sets in Banach spaces”, Ann. of Math. (2) 88 (1968), 3546.
[2] С.м. ГуТман [Gutman, S.M.], “Об эквивалентных нормах в некоторых несепарабедьных в-пространствах” [Equivalent norms in certain nonseparable B–spaces], Teor. Funkciǐ Funkcional. Anal. i Priložen. 20 (1974), 6369.
[3]John, K. and Zizler, V., “Duals of Banach spaces which admit nontrivial smooth functions”, Bull. Austral. Math. Soc. 11 (1974), 161166.
[4]Tacon, D.G., “The conjugate of a smooth Banach space”, Bull. Austral. Math. Soc. 2 (1970), 415425.
[5]Talagrand, Michel, “Renormages de quelques C(K)”, to appear.
[6]Troyanski, S., “On locally uniformly convex and differentiable norms in certain non-separable Banach spaces”, Studia Math. 37 (1971), 173180.
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Locally uniformly rotund renorming and decompositions of Banach spaces

  • V. Zizler (a1)

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