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A survey on Lipschitz-free Banach spaces

100%

Godefroy G.

Commentationes Mathematicae

2015

tom Vol 55, No. 2

89--118

This article is a survey of Lipschitz-free Banach spaces and recent progress in the understanding of their structure.Fe results we present have been obtained in the last fifteen years (and quite often in the last five years). We give a self-contained presentation of the basic properties of Lipschitz-free Banach spaces and investigate some specific topics: non-linear transfer of asymptotic smoothness, approximation properties, norm-attainment. Section 5 consists mainly of unpublished results. A list of open problems with comentary is provided.

A note on differentiability of Lipschitz maps

80%

Górak R.

Bulletin of the Polish Academy of Sciences. Mathematics

2010

tom Vol. 58, no 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 Frechet 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 Gateaux differentiable.

Strong continuity of semigroup homomorphisms

70%

Basit B. , Pryde A.

Studia Mathematica

1999

tom 132

nr 1

71-78

Let J be an abelian topological semigroup and C a subset of a Banach space X. Let L(X) be the space of bounded linear operators on X and Lip(C) the space of Lipschitz functions ⨍: C → C. We exhibit a large class of semigroups J for which every weakly continuous semigroup homomorphism T: J → L(X) is necessarily strongly continuous. Similar results are obtained for weakly continuous homomorphisms T: J → Lip(C) and for strongly measurable homomorphisms T: J → L(X).