2-Local Isometries on Spaces of Lipschitz Functions
Abstract Let ( X , d ) be a metric space, and let Lip( X ) denote the Banach space of all scalar-valued bounded Lipschitz functions ƒ on X endowed with one of the natural norms where L ( ƒ ) is the Lipschitz constant of ƒ . It is said that the isometry group of Lip( X ) is canonical if every surjective linear isometry of Lip( X ) is induced by a surjective isometry of X . In this paper we prove that if X is bounded separable and the isometry group of Lip( X ) is canonical, then every 2-local isometry of Lip( X ) is a surjective linear isometry. Furthermore, we give a complete description of all 2-local isometries of Lip( X ) when X is bounded.