Fouad Chaatit, M. Khamsi
We prove that a Banach lattice $X$ which does not contain the $l^n_{\infty}$-uniformly has an equivalent norm which is uniformly Kadec-Klee for a natural topology $τ$ on $X$. In case the Banach lattice is purely atomic, the topology $τ$ is the coordinatewise convergence topology.
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It is shown that if the dual of a Banach space, $X^*$, where the dual ball is weak* sequentially compact, has the weak* uniform Kadec-Klee property then $X^*$ has Property($K^*$). An example is given where the reverse implication does not hold. That is, there is a Banach space $X$ whose dual, $X^*$, has Property($K^*$) but $X^*$ does not have the weak* uniform Kadec-Klee property.
We show that if a rearrangement invariant Banach function space $E$ on the positive semi-axis satisfies a non-trivial lower $q-$ estimate with constant $1$ then the corresponding space $E(\nm)$ of $τ-$measurable operators, affiliated with an arbitrary semi-finite von Neumann algebra $\nm$ equipped with a distinguished faithful, normal, semi-finite trace $τ$, has the uniform Kadec-Klee property for the topology of local convergence in measure. In particular, the Lorentz function spaces $L_{q,p}$ and the Lorentz-Schatten classes ${\cal C}_{q,p}$ have the UKK property for convergence locally in m…
In this paper, we study the Kadec-Klee property for convergence in measure of noncommutative Orlicz spaces $L_φ(\widetilde{\mathcal{M}},τ)$, where $\widetilde{\mathcal{M}}$ is a von Neumann algebra, and $φ$ is an Orlicz function. We show that if $φ\inΔ_{2}$, $L_φ(\widetilde{\mathcal{M}},τ)$ has the Kadec-Klee property in measure. As a corollary, the dual space and reflexivity of $L_φ(\widetilde{\mathcal{M}},τ)$ are given.
In this paper, we completely settle several of the open questions regarding the relationships between the three most fundamental forms of set convergence. In particular, it is shown that Wijsman and slice convergence coincide precisely when the weak star and norm topologies agree on the dual sphere. Consequently, a weakly compactly generated Banach space admits a dense set of norms for which Wijsman and slice convergence coincide if and only if it is an Asplund space. We also show that Wijsman convergence implies Mosco convergence precisely when the weak star and Mackey topologies coincide on …
We explore new implications of the $M(r,s)$ and $M^*(r,s)$ properties for Banach spaces. We show that a Banach space $X$ satisfying property $M(1,s)$ for some $0<s\leq 1$, admitting a point $x_{0}$ in its unit sphere at which the relative weak and norm topologies agree, satisfies the generalized Gossez-Lami Dozo property. We establish sufficient conditions, in terms of the $(r, s)$-Lipschitz weak$^*$ Kadec-Klee property on a Banach space $X$, to guarantee that its dual space satisfies the UKK$^*$ property. We determine appropriate conditions to assure that a Banach space $X$ satisfies the $(r,…
Concentration theorems for $2$-homogeneous polynomials and bilinear forms with applications to the Kadec-Klee properties
该研究针对p-凸Banach序列格上的2-齐次多项式与双线性型建立集中定理，推导其Kadec-Klee性质，明确最优条件并拓展至算子空间，还确立了相关张量范数的强次可微性。
We research proximinality of $μ$-sequentially compact sets and $μ$-compact sets in measurable function spaces. Next we show a correspondence between the Kadec-Klee property for convergence in measure and $μ$-compactness of the sets in Banach function spaces. Also the property $S$ is investigated in Fréchet spaces and employed to provide the Kadec-Klee property for local convergence in measure. We discuss complete criteria for continuity of metric projection in Fréchet spaces with respect to the Hausdorff distance. Finally, we present the necessary and sufficient condition for continuous metric…
Sphere version of Banach--Kadec--Paley theorem
本文引入几乎Lipschitz嵌入概念，证明单位球几乎Lipschitz嵌入等价于线性嵌入，从而球的几何完全决定L_p空间的线性结构，并给出Hölder几何的锐利结果。
A study on $\mr{F}$-simultaneous approximative $τ$-compactness property in Banach spaces
本文研究了Banach空间中F-同时近似τ-紧性性质，探讨了其与反射性、Fréchet光滑性和Kadec-Klee性质的关系。
Trees are very agreeable objects to work with, offering a diversity of behaviour within a structure that is sufficiently simple to admit precise analysis. Thus we are able to offer fairly satisfactory necessary and sufficient conditions on a tree $Υ$ for the existence of equivalent LUR or strictly convex norms on $\C_0(Υ)$ and for norms with the Kadec Property. In particular, we show that for a {\sl finitely branching} tree $Υ$ the space $\C_0(Υ)$ admits a Kadec renorming. Since some finitely branching trees fail the condition for strictly convex renormability, we obtain an example of a Banach…
A version of the classical Klee-Andô Theorem states the following: For every Banach space $X$, ordered by a closed generating cone $C\subseteq X$, there exists some $α>0$ so that, for every $x\in X$, there exist $x^{\pm}\in C$ so that $x=x^{+}-x^{-}$ and $\|x^{+}\|+\|x^{-}\|\leqα\|x\|$. The conclusion of the Klee-Andô Theorem is what is known as a conormality property. We prove stronger and somewhat more general versions of the Klee-Andô Theorem for both conormality and coadditivity (a property that is intimately related to conormality). A corollary to our result shows that the functions $x\ma…
A typical result in this note is that if $X$ is a Banach space which is a weak Asplund space and has the $ω^*$-$ω$-Kadets Klee property, then $X$ is already an Asplund space.
Lacunary Series, Nonlinear Functionals and Banach Space Structure
研究缺项随机变量序列的非线性泛函的渐近行为，建立Aldous子序列原理的一致版本，并证明Orlicz空间中的Kadec-Pełczyński型定理。
A Banach space has the weak fixed point property if its dual space has a weak$^*$ sequentially compact unit ball and the dual space satisfies the weak$^*$ uniform Kadec-Klee property; and it has the \fpp if there exists $ε>0$ such that, for every infinite subset $A$ of the unit sphere of the dual space, $A\cup (-A)$ fails to be $(2-ε)$-separated. In particular, $E$-convex Banach spaces, a class of spaces that includes the uniformly nonsquare spaces, have the fixed point property.
This paper studies the bounded approximation property (BAP) in quasi Banach spaces. In the first part of the paper we show that the kernel of any surjective operator $\ell_p\to X$ has the BAP when $X$ has it and $0<p\leq 1$, which is an analogue of the corresponding result of Lusky for Banach spaces. We then obtain and study nonlocally convex versions of the Kadec-Pełczyński-Wojtaszczyk complementably universal spaces for Banach spaces with the BAP.
On strengthened versions of Klee's convex body problem in Banach spaces
研究Banach空间中凸体在局部一致凸、一致凸、Fréchet光滑和一致光滑意义下的逼近问题，给出了存在等价范数的充要条件。
Using the game approach to fragmentability, we give new and simpler proofs of the following known results: (a) If the Banach space admits an equivalent Kadec norm, then its weak topology is fragmented by a metric which is stronger than the norm topology. (b) If the Banach space admits an equivalent rotund norm, then its weak topology is fragmented by a metric. (c) If the Banach space is weakly locally uniformly rotund, then its weak topology is fragmented by a metric which is stronger than the norm topology.
Direct sums and abstract Kadets--Klee properties
本研究探究Banach空间直和构造的抽象Kadets--Klee性质，指出仅分量空间具备该性质不足以实现提升，需补充Schur型性质与序列空间严格单调性的二分条件，该机制可复现并改进Köthe--Bochner序列空间的相关经典结果。
We introduce the notion of approximate norm attainment set of a bounded linear operator between Banach spaces and use it to obtain a complete characterization of smooth points in the space of compact linear operators, provided the domain space is reflexive and Kadets-Klee. We also apply the concept to characterize strong BPB property (sBPBp) of a pair of Banach spaces. We further introduce uniform $ ε- $BPB approximation of a bounded linear operator and uniform strong BPB property (uniform sBPBp) with respect to a given family of norm one linear operators and explore some of the relevant prope…
On Klee's problem of convex bodies in Banach spaces
本文解决了Klee于1959年提出的无限维Banach空间中凸体一致逼近问题，给出了严格凸逼近的充要条件，并证明了可分或自反空间中的严格凸且光滑逼近，通过建立局部Lipschitz同构实现。
The $Λ$-sequence spaces $Λ_p$ for $1< p\leq\infty$ and its generalization $Λ_{\hat{p}}$ for $1<\hat{p}<\infty$, $\hat{p}=(p_n)$ is introduced. The James constants and strong $n$-th James constants of $Λ_p$ for $1<p\leq\infty$ is determined. It is proved that generalized $Λ$-sequence space $Λ_{\hat{p}}$ is embedded isometrically in the Nakano sequence space $l_{\hat{p}}(\mathbb{R}^{n+1})$ of finite dimensional Euclidean space $\mathbb{R}^{n+1}$. Hence it follows that sequence spaces $Λ_p$ and $Λ_{\hat{p}}$ possesses the uniform Opial property, property $(β)$ of Rolewicz and weak uniform normal …
Delta-points and their implications for the geometry of Banach spaces
该研究在Banach空间几何领域取得多项进展：构造具Δ-点的超自反空间，证明任意无限维Banach空间可重赋范出Δ-点，建立Δ-点与对偶空间的关联并给出若干不含Δ-点的空间类，还给出Lipschitz-free空间中Δ-点的度量刻画并解决相关公开问题。
A small Banach space $C(K)$ without nice renormings
在 $\omega_1<\mathfrak{c}$ 的一致假设下，构造了一个紧空间 $K$，使得连续函数空间 $C(K)$ 是 Grothendieck 空间、密度为 $\omega_1$，且不存在严格凸或序列 Kadets–Klee 重赋范。
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