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Perturbations of isometries between Banach spaces

Rafał Górak — 2011

Studia Mathematica

We prove a very general theorem concerning the estimation of the expression ||T((a+b)/2) - (Ta+Tb)/2|| for different kinds of maps T satisfying some general perturbed isometry condition. It can be seen as a quantitative generalization of the classical Mazur-Ulam theorem. The estimates improve the existing ones for bi-Lipschitz maps. As a consequence we also obtain a very simple proof of the result of Gevirtz which answers the Hyers-Ulam problem and we prove a non-linear generalization of the Banach-Stone...

A Note on Differentiability of Lipschitz Maps

Rafał Górak — 2010

Bulletin of the Polish Academy of Sciences. Mathematics

We show that every Lipschitz map defined on an open subset of the Banach space C(K), where K is a scattered compactum, with values in a Banach space with the Radon-Nikodym property, has a point of Fréchet differentiability. This is a strengthening of the result of Lindenstrauss and Preiss who proved that for countable compacta. As a consequence of the above and a result of Arvanitakis we prove that Lipschitz functions on certain function spaces are Gâteaux differentiable.

Function spaces on ordinals

Rafał Górak — 2005

Commentationes Mathematicae Universitatis Carolinae

We give a partial classification of spaces C p ( [ 1 , α ] ) of continuous real valued functions on ordinals with the topology of pointwise convergence with respect to homeomorphisms and uniform homeomorphisms.

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