M.Sc Thesis | |

M.Sc Student | Einav Amit |
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Subject | Lipschitz Maps and Lipschitz Functions |

Department | Department 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 Lip_{0}(M)
and lip_{0}(M) and generalize known results on their structure.

The Thesis is mainly based on recent papers by G. Godefroy
and N. J. Kalton.