L'espace de Banach C0 est déterminé par sa métrique

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In this Note we show that a Banach space which is Lipschitz-isomorphic to a subspace of c 0 is linearly isomorphic to a subspace of c 0. We deduce that a space which is Lipschitz-isomorphic to c 0 is in fact linearly isomorphic to c 0. We also prove that a space which is uniformly homeomorphic to a subspace of c 0 has a summable Szlenk index. Finally, we investigate the extension of these results to the non-separable case.