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- W4299387529 abstract "Given Banach spaces E and F, we denote by ${mathcal P}(^k!E,F)$ the space of all k-homogeneous (continuous) polynomials from E into F, and by ${mathcal P}_{wb}(^k!E,F)$ the subspace of polynomials which are weak-to-norm continuous on bounded sets. It is shown that if E has an unconditional finite dimensional expansion of the identity, the following assertions are equivalent: (a) ${mathcal P}(^k!E,F)={mathcal P}_{wb}(^k!E,F)$; (b) ${mathcal P}_{wb}(^k!E,F)$ contains no copy of $c_0$; (c) ${mathcal P}(^k!E,F)$ contains no copy of $ell_infty$; (d) ${mathcal P}_{wb}(^k!E,F)$ is complemented in ${mathcal P}(^k!E,F)$. This result was obtained by Kalton for linear operators. As an application, we show that if E has Pel czy'nski's property (V) and satisfies ${mathcal P}(^k!E) ={mathcal P}_{wb}(^k!E)$ then, for all F, every unconditionally converging $Pin{mathcal P}(^k!E,F)$ is weakly compact. If E has an unconditional finite dimensional expansion of the identity, then the converse is also true." @default.
- W4299387529 created "2022-10-02" @default.
- W4299387529 creator A5005953479 @default.
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- W4299387529 date "2000-03-18" @default.
- W4299387529 modified "2023-10-13" @default.
- W4299387529 title "The polynomial property (V)" @default.
- W4299387529 doi "https://doi.org/10.48550/arxiv.math/0003111" @default.
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