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arXiv 1407.7599math.FA
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Lipschitz函数空间的双对偶性与稠密性

Biduality and density in Lipschitz function spaces

A. Jiménez-Vargas, J. M. Sepulcre, Moisés Villegas-Vallecillos

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7 pages
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摘要：

对于点紧度量空间$(X,d)$，我们讨论了Lipschitz函数空间$\mathrm{Lip}_0(X,d)$与小Lipschitz函数空间$\mathrm{lip}_0(X,d)$的双对偶何时同构的问题，并证明了只要$\mathrm{lip}_0(X,d)$的闭单位球在$\mathrm{Lip}_0(X,d)$的闭单位球中关于点式收敛拓扑是稠密的，这就是这种情况.然后，我们应用我们的密度准则，以另一种方式证明了一个经典的结果，断言$\mathrm{Lip}_0(X,d^α)$是等距同构$\mathrm{lip}_0(X,d^α)^{**}$的任何$α$在$(0,1)$的真实版本。

英文摘要：

For pointed compact metric spaces $(X,d)$, we address the biduality problem as to when the space of Lipschitz functions $\mathrm{Lip}_0(X,d)$ is isometrically isomorphic to the bidual of the space of little Lipschitz functions $\mathrm{lip}_0(X,d)$, and show that this is the case whenever the closed unit ball of $\mathrm{lip}_0(X,d)$ is dense in the closed unit ball of $\mathrm{Lip}_0(X,d)$ with respect to the topology of pointwise convergence. Then we apply our density criterion to prove in an alternate way the real version of a classical result which asserts that $\mathrm{Lip}_0(X,d^α)$ is isometrically isomorphic to $\mathrm{lip}_0(X,d^α)^{**}$ for any $α$ in $(0,1)$.

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