We show that when a linear quotient map to a separable Banach space X has a Lipschitz right inverse, then it has a linear right inverse. If a separable space X embeds isometrically into a Banach space Y , then Y contains an isometric linear copy of X. This is false for every nonseparable weakly compactly generated Banach space X. Canonical examples of nonseparable Banach spaces which are Lipschitz isomorphic but not linearly isomorphic are constructed. If a Banach space X has the bounded approximation property and Y is Lipschitz isomorphic to X, then Y has the bounded approximation property. 1. Introduction. A (real) Banach space X is in particular a metric space, equipped with the distinguished point f0g, and to such a pointed metric space we can associate the space Lip 0 (X) of all real-valued Lipschitz functions which vanish at 0. We refer to (29) for basic facts on this space and some of its uses. It is clear that Lip 0 (X) is a Banach space when it is equipped with the Lipschitz norm, dened

https://www.impan.pl/shop/publication/transaction/download/product/90209?download.pdf

Lipschitz-free Banach spaces

We introduce the notion of cotype of a metric space, and prove that for Banach spaces it coincides with the classical notion of Rademacher cotype This yields a concrete version of Ribe's theorem, settling a long standing open problem in the nonlinear theory of Banach spaces We apply our results to several problems in metric geometry Namely, we use metric cotype in the study of uniform and coarse embeddings, settling in particular the problem of classifying when L_p coarsely or uniformly embeds into L_q We also prove a nonlinear analog of the Maurey-Pisier theorem, and use it to answer a question posed by Arora, Lovasz, Newman, Rabani, Rabinovich and Vempala, and to obtain quantitative bounds in a metric Ramsey theorem due to Matousek

Metric Cotype

/pdf/on-uniformly-homeomorphic-normed-spaces-ii-4yhylg3oh1.pdf

On uniformly homeomorphic normed spaces II

It is proved using positive definite functions that a normed spaceX is unifomly homeomorphic to a subset of a Hilbert space, if and only ifX is (linearly) isomorphic to a subspace of aL
 0(μ) space (=the space of the measurable functions on a probability space with convergence in probability). As a result we get thatl
 p
 (respectivelyL
 p
 (0, 1)), 2<p<∞, is not uniformly embedded in a bounded subset of itself. This answers negatively the question whether every infinite dimensional Banach space is uniformly homeomorphic to a bounded subset of itself. Positive definite functions are also used to characterize geometrical properties of Banach spaces.

Uniform embeddings of metric spaces and of banach spaces into hilbert spaces

Geometric embeddings of metric spaces

It is shown that there is a constantK so that, for every separable metric spaceX, there is a mapT:X →co satisfyingd(x, y)≦‖Tx−Ty‖≦Kd(x, y) for everyx, y ∈ X.

Every separable metric space is Lipschitz equivalent to a subset ofc0

It is proved that for 1<-p≤2,Lp(0,1) andlp are uniformly equivalent to bounded subsets of themselves. It is also shown that for 1<=p<=2, 1≦q<∞,Lp is uniformly equivalent to a subset oflq.

Israel Aharoni

Papers

Every separable metric space is Lipschitz equivalent to a subset ofc0

Uniform embeddings of Banach spaces