Let $\mathrm {Lip}_0(M)$ be the space of Lipschitz functions on a complete metric space M that vanish at a base point. We prove that every normal functional in ${\mathrm {Lip}_0(M)}^*$ is weak* continuous; that is, in order to verify weak* continuity it suffices to do so for bounded monotone nets of Lipschitz functions. This solves a problem posed by N. Weaver. As an auxiliary result, we show that the series decomposition developed by N. J. Kalton for functionals in the predual of $\mathrm {Lip}_0(M)$ can be partially extended to ${\mathrm {Lip}_0(M)}^*$ .

中文翻译：

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LIPSCHITZ 空间上的正规泛函弱*连续

设 $\mathrm {Lip}_0(M)$ 是完全度量空间M上的 Lipschitz 函数空间，该空间在基点处消失。我们证明 ${\mathrm {Lip}_0(M)}^*$ 中的每个正规泛函都是弱*连续的；也就是说，为了验证弱*连续性，对 Lipschitz 函数的有界单调网这样做就足够了。这解决了 N. Weaver 提出的问题。作为辅助结果，我们表明 NJ Kalton 为 $\mathrm {Lip}_0(M)$ 的预对偶中的泛函开发的级数分解可以部分扩展到 ${\mathrm {Lip}_0(M)}^ *$ 。