M.Sc Thesis

M.Sc StudentEinav Amit
SubjectLipschitz Maps and Lipschitz Functions
DepartmentDepartment of Mathematics
Supervisor PROFESSOR EMERITUS Yoav Benyamini


We investigate the lifting property for Banach spaces. A Banach space X is said to have the lifting property if for every Banach space Y and every linear quotient map Q from Y to X which has a Lipschitz lifting also has a linear lifting.
For every Banach space X we associate a new Banach space F(X), the free space of X, and a quotient map β from F(X) to X with a natural Lipschitz lifting. We show that F(X) is Lipschitz isomorphic to the direct sum of Ker (β) and X, and the spaces are linearly isomorphic if and only if X has the lifting property.
We prove that a non-separable WCG space does not have the lifting property. This gives natural examples of pairs of Lipschitz isomorphic Banach spaces which are not linearly isomorphic. These examples generalize earlier ad-hoc constructions by Aharoni-Lindenstrauss, and Deville, Godefroy & Zizler.
We also prove that every separable Banach space has the lifting property. Thus the question whether there exist two separable Banach spaces which are Lipschitz isomorphic yet not linearly isomorphic remains open.
We also use similar ideas to construct Banach spaces that are uniformly homeomorphic yet are not linearly isomorphic. Earlier examples by Ribe and Aharoni-Lindenstrauss used completely different methods.
In conclusion we investigate the relationship between Lip0(M) and lip0(M) and generalize known results on their structure.
The Thesis is mainly based on recent papers by G. Godefroy and N. J. Kalton.