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Linearly rigid metric spaces. (Espaces métriques linéairement rigides.) (French) Zbl 1121.54043

This paper studies complete separable metric spaces \((X,d)\) which admit a unique linearly dense embedding into a Banach space; such spaces are called linearly rigid metric spaces. The notion goes back to Kantorovich and Banach who gave the construction of such an embedding. Choosing \(x_0 \in X\) they defined the Banach space \(E(X,d,x_0)\) into which the space \((X,d)\) is isometrically embedded in such a way that the linear span of its image is dense in \(E(X,d,x_0)\) and the point \(x_0\) is mapped into the zero point. The authors also characterize those complete separable metric spaces which do not admit other isometrical embeddings of this kind except those given by the Kantorovich-Banach construction.

MSC:

54E35 Metric spaces, metrizability
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