An extension of a result of ribe

Aharoni, Israel; Lindenstrauss, Joram

doi:10.1007/bf02776080

Cited by 13 publications

(7 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For X = ( ⊕ℓ p n ) q and p n → p, this theorem was proved by M. Ribe [Rib3] for p = 1 and extended by I. Aharoni and J. Lindenstrauss [AL2] for p > 1. The proof we sketch closely follows a simplification of the proof in [AL2] given by Y. Benyamini in his nice exposition [Ben2].…”

Section: Uniform Homeomorphisms Between X and ℓ P ⊕ Xmentioning

confidence: 88%

“…. .. From the construction of [AL2] one easily gets examples of spaces whose uniform structure determine 2 ℵ 0 isomorphism classes. If α is any cardinal less than the continuum which is not a power of two, in particular if α = 3 or α = ℵ 0 , we do not know how to construct a space which determines exactly α isomorphism classes.…”

Section: Spaces Determined By Their Finite Dimensional Subspacesmentioning

confidence: 99%

See 1 more Smart Citation

Banach spaces determined by their uniform structures

Johson

Lindenstrauss

Schechtman

1996

Geometric and Functional Analysis

View full text Add to dashboard Cite

Following results of Bourgain and Gorelik we show that the spaces gp, 1 < p < oo, as well as some related spaces have the following uniqueness property: If X is a Bana~h space uniformly homeomorphic to one of these spaces then it is linearly isomorphic to the same space. We also prove that if a C(K) space is uniformly homeomorphic to c0, then it is isomorphic to Co. We show also that there are Banach spaces which are uniformly homeomorphic to exactly 2 isomorphically distinct spaces.

show abstract

Section: Uniform Homeomorphisms Between X and ℓ P ⊕ Xmentioning

confidence: 88%

Section: Spaces Determined By Their Finite Dimensional Subspacesmentioning

confidence: 99%

Banach spaces determined by their uniform structures

Johson

Lindenstrauss

Schechtman

1996

Geometric and Functional Analysis

View full text Add to dashboard Cite

show abstract

“…A Banach space X has the alternating p-Banach-Saks property (respectively weak p-Banach-Saks property) if it has the alternating p-Banach-Saks (respectively weak p-Banach-Saks) property with constant C for some C > 0. 3 A Banach space has the alternating p-Banach-Saks property if and only if it does not contain 1 and it has the weak p-Banach-Saks property (cf. [8,Proposition 3.1]).…”

Section: Banach-saks Properties and Asymptotic Uniform Smoothnessmentioning

confidence: 99%

On Asymptotic Uniform Smoothness and Nonlinear Geometry of Banach Spaces

Braga¹

2019

J. Inst. Math. Jussieu

View full text Add to dashboard Cite

These notes concern the nonlinear geometry of Banach spaces, asymptotic uniform smoothness and several Banach-Saks-like properties. We study the existence of certain concentration inequalities in asymptotically uniformly smooth Banach spaces as well as weakly sequentially continuous coarse (Lipschitz) embeddings into those spaces. Some results concerning the descriptive set theoretical complexity of those properties are also obtained. We finish the paper with a list of open problem.

show abstract

“…the direct sum is taken in the ℓ 1 norm). In [AL2] the argument of Ribe was modified so that it works also if 1 is replaced by s; 1 < s < p and thus one gets even a superreflexive and separable example. A perhaps more striking example is presented in [JLS].…”

Section: Uniform and Lipschitz Classification Of Spaces And Ballsmentioning

confidence: 99%

Uniform embeddings, homeomorphisms and quotient maps between Banach spaces (a short survey)

Lindenstrauss

1998

Topology and its Applications

Self Cite

View full text Add to dashboard Cite

ii) Uniform and Lipschitz classification of Banach spaces and their balls.(iii) Uniform and Lipschitz quotient maps. I will just state the main results, explain them and give references to the papers in which they were originally proved. For complete proofs, additional results and further references I refer to the forthcoming book.It is worthwhile to mention that the embedding problems treated in (i) above have discrete analogues which lead to the study of natural problems on finite metric spaces (usually graphs with their obvious metric). These problems are of a combinatorial nature and are connected to topics in computer science. This direction is however not discussed here and again I refer to [BL] for a detailed treatment of this topic.The theory of Lipschitz and uniformly continuous functions on Banach spaces has been developing in a slow but rather steady pace over the last 35 years and by now much is known in this direction. Nevertheless many basic and natural questions remain unanswered. In the last section of this paper I present a sample of open problems (not necessarily the central ones) which are related to the material discussed in the other sections.Finally let me mention that the recent introductory text on Banach space theory [HHZ] contains in its last chapter (chapter 12) an introduction (with proofs) to the subject matter of the present survey. EmbeddingsLet us start by examining Lipschitz embeddings. The main question here is the following: Assume that there is a Lipschitz embedding f from a Banach space X into a Banach space Y (i.e. f is a Lipschitz injection and f −1 is also Lipschitz on its domain of definition). Does this imply that X is actually linearly isomorphic to a subspace of Y ?The main tool for handling this problem is differentiation. Let us recall the definition of the two main types of differentiation. A map f defined on open set G in a Banach space X into a Banach space Y is called Gâteaux differentiable at x 0 ∈ G if for every u ∈ X ( * ) lim t→∞ (f (x 0 + tu) − f (x 0 ))/t = D f (x 0 )u of a.e. (and the notions are definitely not equivalent) but all will suit us here. For exampleChristensen calls a Borel set A a null set if there is a Radon probability measure µ on X so that µ(A + x) = 0 for all x ∈ X. It is easy to see that if dim X < ∞ then A is null in the above sense iff A is of Lesbegue measure 0. It is also not hard to verify that a countable union of null sets is a null set and that a null set has empty interior.An immediate corollary of the theorem is the following statement.Assume X is separable and that there is a Lipschitz embedding of it into a space Y with RNP. Then there is a linear isomorphism from X into Y .What happens if Y fails to have RNP? Let us check first the case Y = c 0 . The following result was proved by Aharoni [Ah1].Every separable Banach space is Lipschitz equivalent to a subset of c 0 .Recall that c 0 is a "small" space, actually a minimal space in the following sense.Any infinite dimensional subspace of c 0 has in turn a subspace isomorphic ...

show abstract

An extension of a result of ribe

Cited by 13 publications

References 3 publications

Banach spaces determined by their uniform structures

Banach spaces determined by their uniform structures

On Asymptotic Uniform Smoothness and Nonlinear Geometry of Banach Spaces

Uniform embeddings, homeomorphisms and quotient maps between Banach spaces (a short survey)

Contact Info

Product

Resources

About