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Approximation of Lipschitz functions by \(\Delta\)-convex functions in Banach spaces. (English) Zbl 0920.46010

Summary: We give some results about the approximation of a Lipschitz function on a Banach space by means of \(\Delta\)-convex functions. In particular, we prove that the density of \(\Delta\)-convex functions in the set of Lipschitz functions for the topology of uniform convergence on bounded sets characterizes the superreflexivity of the Banach space. We also show that Lipschitz functions on superreflexive Banach spaces are uniform limits on the whole space of \(\Delta\)-convex functions.

MSC:

46B20 Geometry and structure of normed linear spaces
46B10 Duality and reflexivity in normed linear and Banach spaces
41A30 Approximation by other special function classes
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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