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A Note on Differentiability of Lipschitz Maps

100%

Górak R.

Bulletin of the Polish Academy of Sciences. Mathematics

2010

tom 58

nr 3

259-268

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.

Perturbations of isometries between Banach spaces

100%

Górak R.

Studia Mathematica

2011

tom 207

nr 1

47-58

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 theorem which improves the results of Jarosz and more recent results of Dutrieux and Kalton.

Ograniczanie wyników

1 Bulletin of the Polish Academy of Sciences. Mathematics

1 Studia Mathematica

2 Górak R.

1 2011

1 2010

Wyniki wyszukiwania

A Note on Differentiability of Lipschitz Maps

Perturbations of isometries between Banach spaces