We give a characterization of uniformly convex Banach spaces in terms of a uniform version of the Kadec-Klee property. As an application we prove that if (xn) is a bounded sequence in a uniformly convex Banach space X which is ε-separated for some 0 < ε ≤ 2, then for all norm one vectors x ∈ X there exists a subsequence (xn j) of (xn) such that inf j =k x − (xn j − xn k) ≥ 1 + δ X (2 3 ε), wher...

The trigonometric interpolants to a periodic function f in equispaced points converge if f is Dini-continuous, and the associated quadrature formula, the trapezoidal rule, converges if f is continuous. What if the points are perturbed? With equispaced grid spacing h, let each point be perturbed by an arbitrary amount ≤ αh, where α ∈ [0, 1/2) is a fixed constant. The Kadec 1/4 theorem of samplin...

We say that the quasi-norm Φ of a symmetric sequence space E has property P if, for all x, y ∈ E with 0 ≤ xn ≤ yn for all n, and xN < yN for some N , we have Φ (xn) < Φ (yn) . Simon [6] proved that if the Banach norm Φ has property P , then whenever An → A in the weak operator topology on the unitary matrix space CE and Φ(An)→ Φ(A), then An → A in the strong operator topology on CE . In this no...

* Correspondence: changss@yahoo. cn; [email protected] Department of Mathematics, Yibin university, Yibin, Sichuan 644007, China Department of Mathematics, College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China Full list of author information is available at the end of the article Abstract The purpose of this article is to introduc...

Let X be a real strictly convex and Kadec Banach space and G a nonempty closed relatively boundedly weakly compact subset of X : Let BðX Þ (resp. KðXÞ) be the family of nonempty bounded closed (resp. compact) subsets of X endowed with the Hausdorff distance and let BGðXÞ denote the closure of the set fAABðX Þ : A-G 1⁄4 |g and KGðX Þ 1⁄4 BGðX Þ-KðXÞ: We introduce the admissible family A of BðX Þ...

Given a finite number of moments of an unknown density on a finite measure space, the best entropy estimate-that nonnegative density x with the given moments which minimizes the Boltzmann-Shannon entropy I(x):= x log x-is considered. A direct proof is given that I has the Kadec property in L-if Yn converges weakly to 37 and I(yn) converges to I(37), then yn converges to 37 in norm. As a corolla...

Contents 1. Introduction 2 2. Metric type 7 2.1. Type for metric spaces 8 2.2. The geometric puzzle 11 2.3. Pisier's argument 12 2.4. Unique obstructions to type 17 3. Metric cotype 19 4. Markov type and cotype 22 4.1. Lipschitz extension via Markov type and cotype 27 5. Markov convexity 29 6. Metric smoothness? 32 7. Bourgain's discretization problem 32 8. Nonlinear Dvoretzky theorems 33 8.1. ...

Abstract The aim of this note is to obtain results about when the norm a projective tensor product strongly subdifferentiable. We prove that if $$X\widehat{\otimes }_\pi Y$$ X ⊗ ^ π Y ...