Both classical linear and multilinear isometries defined between subalgebras of bounded continuous functions on (complete) metric spaces are studied. Particularly, we prove that certain such subalgebras, including the subalgebras of uniformly continuous, Lipschitz or locally Lipschitz functions, determine the topology of (complete) metric spaces. As a consequence, it is proved that the subalgebra of Lipschitz functions determines the Lipschitz in the small structure of a complete metric space. Furthermore, we provide a weighted composition representation for multilinear isometries from similar subalgebras on (not necessarily complete) metric spaces. We apply this general representation to obtain more specific ones for subalgebras of uniformly continuous and Lipschitz functions.

研究了（完整）度量空间上有界连续函数的子代数之间定义的经典线性和多线性等式。特别是，我们证明了某些此类子代数，包括一致连续的Lipschitz函数或局部Lipschitz函数的子代数，确定了（完整）度量空间的拓扑。结果证明，Lipschitz函数的子代数决定了完整度量空间的小结构中的Lipschitz。此外，我们为度量空间上的（不一定是完整的）相似子代数的多线性等式提供了加权组合表示。我们应用这种一般表示来获得统一连续和Lipschitz函数的子代数的更具体的表示。