Bourgain’s discretization theorem asserts that there exists a universal constant $C\in (0,\infty )$ with the following property. Let $X,Y$ be Banach spaces with $dimX=n$. Fix $D\in (1,\infty )$ and set $\delta =e^{-n^{Cn}}$. Assume that $𝒩$ is a $\delta$-net in the unit ball of $X$ and that $𝒩$ admits a bi-Lipschitz embedding into $Y$ with distortion at most $D$. Then the entire space $X$ admits a bi-Lipschitz embedding into $Y$ with distortion at most $CD$. This mostly expository article is devoted to a detailed presentation of a proof of Bourgain’s theorem.We also obtain an improvement...

In this paper we study the notions of finite turn of a curve and finite turn of tangents of a curve. We generalize the theory (previously developed by Alexandrov, Pogorelov, and Reshetnyak) of angular turn in Euclidean spaces to curves with values in arbitrary Banach spaces. In particular, we manage to prove the equality of angular turn and angular turn of tangents in Hilbert spaces. One of the implications was only proved in the finite dimensional context previously, and equivalence of finiteness...

We introduce and investigate the non-n-linear concept of fully summing mappings; if n = 1 this concept coincides with the notion of nonlinear absolutely summing mappings and in this sense this article unifies these two theories. We also introduce a non-n-linear definition of Hilbert-Schmidt mappings and sketch connections between this concept and fully summing mappings.

We give an example of a uniform quotient map from R2 to R which has non-locally connected level sets.

We show that there is no uniformly continuous selection of the quotient map $Q:{\ell }_{\infty }\to {\ell }_{\infty }/c₀$ relative to the unit ball. We use this to construct an answer to a problem of Benyamini and Lindenstrauss; there is a Banach space X such that there is a no Lipschitz retraction of X** onto X; in fact there is no uniformly continuous retraction from $B_{X**}$ onto $B_X$.

We show that the Hilbert space is coarsely embeddable into any ${\ell }_p$ for 1 ≤ p ≤ ∞. It follows that coarse embeddability into ℓ₂ and into ${\ell }_p$ are equivalent for 1 ≤ p < 2.

We study the Gromov-Hausdorff and Kadets distances between C(K)-spaces and their quotients. We prove that if the Gromov-Hausdorff distance between C(K) and C(L) is less than 1/16 then K and L are homeomorphic. If the Kadets distance is less than one, and K and L are metrizable, then C(K) and C(L) are linearly isomorphic. For K and L countable, if C(L) has a subquotient which is close enough to C(K) in the Gromov-Hausdorff sense then K is homeomorphic to a clopen subset of L.

Sobczyk's theorem asserts that every c₀-valued operator defined on a separable Banach space can be extended to every separable superspace. This paper is devoted to obtaining the most general vector valued version of the theorem, extending and completing previous results of Rosenthal, Johnson-Oikhberg and Cabello. Our approach is homological and nonlinear, transforming the problem of extension of operators into the problem of approximating z-linear maps by linear maps.

The purpose of this paper is to establish some common fixed point results for $f$-nondecreasing mappings which satisfy some nonlinear contractions of rational type in the framework of metric spaces endowed with a partial order. Also, as a consequence, a result of integral type for such class of mappings is obtained. The proved results generalize and extend some of the results of J. Harjani, B. Lopez, K. Sadarangani (2010) and D. S. Jaggi (1977).