可测全准序空间的最小元和a.e.最小元

本文在拓扑空间、由序诱导的拓扑空间和测度空间中研究可测全准序空间的最小元a.e.最小元和W_(ij)点序列的存在性以及它们之间的相互关系等问题,得到一系列的重要定理。     

四元数Hilbert空间中Riesz基的刻画*

四元数Hilbert空间在应用物理科学特别是量子物理中占有重要地位.本文讨论四元数Hilbert空间的框架理论, 在四元数Hilbert空间中引入了Riesz基的概念, 在此基础上刻画了Riesz基,给出了它们的一些等价条件; 特别地, 得到了四元数Hilbert空间中的一个序列是Riesz基的充要条件是它是一个具有双正交序列的完备Bessel序列,且它的双正交序列也是一个完备Bessel序列; 并进一步证明了双正交序列中一个序列的完备性可以从特征刻画中去除.文中举例说明了双正交性、完备性和Bessel性质之间的关系.     

关于完备可分距离空间中复合随机场弱收敛条件的讨论

本文主要讨论了完备可分距离空间中两个复合随机场序列ξε（υε）和ξ０（υ０）之间的关系，得出的条件，并作为应用，本文还顺便讨论了两个有关的例子．     

套代数的不变算子值域格

<正> 设是可分无限维复Hilbert空间上的一个完备的全序子空间格,即的一个完备子空间套(nest),()的子代数Alg={A∈();LatA}称为生成的套代数(nest algebra),记为N.显然,N的不变子空间格恰是,     

极大代数上多输入多输出线性系统的最小实现

研究了极大代数上无穷阵序列{Gi}0∞的实现问题.首先给出了周期阵序列的概念,得到了1-阶和2-阶周期阵序列{Gi}0∞元素之间的关系,并得到多输入多输出情况下线性系统存在1维和2维最小实现的充要条件.     

一致覆盖和度量空间的紧映象

本文利用一致覆盖的概念,讨论了度量空间的序列覆盖紧映象的结构.主要结果有: (1)空间X是局部可分度量空间的序列覆盖紧映象当且仅当X具有由cosmic子空间构成的一致sn网; (2)空间X是局部可分度量空间的序列覆盖,商紧映象当且仅当X是度量空间的序列覆盖,商紧映象且是局部cosmic空间.     

Banach空间常微分方程的解

在[4]中,S.V.Du和V.Lakshmikantham利用紧型条件证明了必存在单调序列{v_n}和{w_n},一致收敛于Banach空间常微分方程初值问题u′=f(t,u),u(0)=u_0的最大解和最小解。在本文中我们证明了,如果Banach空间是弱序列完备的,则[4]中的紧型条件(这是[4]中的一个主要条件)是可以删掉的。我们还对Banach空间周期边值问题证明了类似的结果。     

距离空间上的几乎处处中心极限定理

该文得到了关于一般可分距离空间上独立随机元序列的几乎处处中心极限定理(almost sure central limit theory, 简记为ASCLT). 作为应用, 该文给出了取值于可分Banach空间上随机元序列以及一类随机场序列满足ASCLT的充分条件,最后给出了关于多维随机变量序列极值的ASCLT.     

关于序列覆盖s映射的注记

本文分别给出局部可分度量空间的强序列覆盖（1序列覆盖,2序列覆盖）。映象的新刻画,还分别给出拓扑空间是局部可分度量空间的序列覆盖（紧覆盖）s映象的一个充分条件．     

星可数κ网与序列覆盖S映射

证明了在空间具有星可数κ网的条件下,度量空间的1(2)序列覆盖s映象是局部可分度量空间的1(2)序列覆盖、紧覆盖s映象.     

Some Characterizations of Hereditarily Indecomposable Banach Spaces

In this paper we first discuss the relations between some G-M-type spaces, and the previous eight kinds of G-M-type Banach spaces are merged into four different kinds. Then we build a Generalized Operator Extension Theorem, and introduce the concept of complete minimal sequences. Some sufficient and necessary conditions under which a Banach space is a hereditarily indecomposable space are given. Finally, we give some characterizations of hereditarily indecomposable Banach Spaces.     

Banach spaces without minimal subspaces

We prove three new dichotomies for Banach spaces à la W.T. Gowers' dichotomies. The three dichotomies characterise respectively the spaces having no minimal subspaces, having no subsequentially minimal basic sequences, and having no subspaces crudely finitely representable in all of their subspaces. We subsequently use these results to make progress on Gowers' program of classifying Banach spaces by finding characteristic spaces present in every space. Also, the results are used to embed any partial order of size ℵ₁ into the subspaces of any space without a minimal subspace ordered by isomorphic embeddability.     

Characterizing R-duality in Banach spaces

R-duals of certain sequences in Hilbert spaces were introduced by Casazza, Kutyniok and Lammers in 2004 and later generalized to Banach spaces by Xiao and Zhu. In this paper we provide some characterizations of R-dual sequences in Banach spaces.     

On Minimal Subspaces in Tensor Representations

In this paper we introduce and develop the notion of minimal subspaces in the framework of algebraic and topological tensor product spaces. This mathematical structure arises in a natural way in the study of tensor representations. We use minimal subspaces to prove the existence of a best approximation, for any element in a Banach tensor space, by means of a tensor given in a typical representation format (Tucker, hierarchical, or tensor train). We show that this result holds in a tensor Banach space with a norm stronger than the injective norm and in an intersection of finitely many Banach tensor spaces satisfying some additional conditions. Examples using topological tensor products of standard Sobolev spaces are given.     

An estimate of the Banach-Mazur distances between Hilbert spaces and Banach spaces

We discuss a geometric property that provides tools for the estimate of the Δ₂ on spaces of Rademacher functions with values on a Banach space for particular sequences of vectors. We obtain in this way a technique to study those Banach spaces that are isomorphic to Hilbert spaces. In particular, an estimate of the Banach-Mazur distance between such these spaces is obtained.     

Minimal vectors in arbitrary Banach spaces

We extend the method of minimal vectors to arbitrary Banach spaces. It is proved, by a variant of the method, that certain quasinilpotent operators on arbitrary Banach spaces have hyperinvariant subspaces.

     

Banach空间中脉冲积分-微分方程的迭代解

利用单调迭代技术,本文首先讨论了Banach空间一阶脉冲积分-微分方程初值问题最大解与最小解的存在性.在此基础上,讨论了右端项中带有一阶导数的二阶脉冲积分-微分方程初值问题最大解与最小解的存在性.最后的例子说明对导数的限制条件是可验证的.     

非连续弱紧增算子的不动点及其对Banach空间初值问题的应用

证明了Banach空间上一类非连续的弱紧增算子的不动点定理,特别获得了最大不动点与最小不动点的存在性,改进了已有的某些结果.作为应用,讨论了Banach空间中含间断项的常微分方程初值问题最大解与最小解的存在性.     

On shrinking and boundedly complete Schauder frames of Banach spaces

This paper studies Schauder frames in Banach spaces, a concept which is a natural generalization of frames in Hilbert spaces and Schauder bases in Banach spaces. The associated minimal and maximal spaces are introduced, as are shrinking and boundedly complete Schauder frames. Our main results extend the classical duality theorems on bases to the situation of Schauder frames. In particular, we will generalize James' results on shrinking and boundedly complete bases to frames. Secondly we will extend his characterization of the reflexivity of spaces with unconditional bases to spaces with unconditional frames.     

Remarks on Minimal Sets for Cyclic Mappings in Uniformly Convex Banach Spaces

In this article, we prove that every nonempty and convex pair of subsets of uniformly convex in every direction Banach spaces has the proximal normal structure and then we present a best proximity point theorem for cyclic relatively nonexpansive mappings in such spaces. We also study the structure of minimal sets of cyclic relatively nonexpansive mappings and obtain the existence results of best proximity points for cyclic mappings using some new geometric notions on minimal sets. Finally, we prove a best proximity point theorem for a new class of cyclic contraction-type mappings in the setting of uniformly convex Banach spaces and so, we improve the main conclusions of Eldred and Veeramani.