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New concepts related to approximating a Lipschitz function between Banach spaces by aane functions are introduced. Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the spaces X, Y for which any Lipschitz function from X to Y can be so approximated is obtained. This is applied to the… (More)

We give several sufficient conditions on a pair of Banach spaces X and Y under which each Lipschitz mapping from a domain in X to Y has, for every ǫ > 0, a point of ǫ-Fréchet differentiability. Most of these conditions are stated in terms of the moduli of asymptotic smoothness and convexity, notions which appeared in the literature under a variety of names.… (More)

We give several suucient conditions on a pair of Banach spaces X and Y under which each Lipschitz mapping from a domain in X to Y has, for every > 0, a point of-Fr echet diierentiability. Most of these conditions are stated in terms of the moduli of asymptotic smoothness and convexity, notions which appeared in the literature under a variety of names. We… (More)

In the theory of Banach spaces a rather small class of spaces has always played a central role (actually even before the formulation of the general theory). This class —the class of classical Banach spaces— contains the L p (p) spaces (p a measure , 1 < p < °°) and the C(K) spaces (K compact Hausdorff) and some related spaces. These spaces are very… (More)

It is shown that if f is a mapping of the plane onto itself that is uniformly continuous with modulus of continuity (r) which is o(p r) as r ! 0 and f is also co-uniformly continuous then f = P h where h is a homeomorphism of the plane and P is a complex polynomial. The same conclusion holds also under other assumptions on the moduli of uniform and… (More)