Quasi‐isometric embeddings of C0(K,X)C₀(K, X) spaces which induce isometries whenever XX is a Hilbert space

Suppose that and are locally compact Hausdorff spaces and is a Hilbert space. It is proven that if there exist real numbers , and a map from to satisfying for every and in , then there are a locally compact subset of and a proper continuous function of onto , on the assumption that In this case, as an immediate consequence, generates a linear isometry of into . Even in the Lipschitz case (), this result is the first nonlinear vector generalization of a classical Jarosz theorem (1984) concerning the into linear isomorphisms of spaces of continuous functions on locally compact Hausdorff spaces.