For a compact metric space $(K,\rho)$, the predual of $Lip(K,\rho)$ can be identified with the normed space $M(K)$ of finite (signed) Borel measures on $K$ equipped with the Kantorovich–Rubinstein norm, this is due to Kantorovich [20]. Here we deduce atomic decomposition of $\mathcal M(K)$ by mean of some results from [10]. It is also known, under suitable assumption, that there is a natural isometric isomorphism between $ Lip(K,\rho)$ and $(lip(K, \rho))^{**}$ [15]. In this work we also show that the pair $(lip(K,\rho),Lip(K,\rho))$ can be framed in the theory of $o–O$ type structures introduced by K. M. Perfekt.

中文翻译：

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Lipschitz–Hölder空间中的对偶和距离公式

对于紧凑的度量空间$（K，\ rho）$，可以用$ K $上有限（有符号）Borel度量的范数空间$ M（K）$来标识$ Lip（K，\ rho）$的前导数。配备了Kantorovich–Rubinstein范数，这是由于Kantorovich [20]。在这里，我们根据[10]的一些结果推导出$ \ M（K）$的原子分解。在适当的假设下，还已知在$ Lip（K，\ rho）$和$（lip（K，\ rho））^ {**} $之间存在自然的等距同构[15]。在这项工作中，我们还证明了$（lip（K，\ rho），Lip（K，\ rho））$对可以由KM Perfekt引入的$ o–O $类型结构的理论来构架。