Continuous operators on asymmetric normed spaces

If (X, p) and (Y, q) are two asymmetric normed spaces, the set LC(X, Y) of all continuous linear mappings from (X, p) to (Y, q) is not necessarily a linear space, it is a cone. If X and Y are two Banach lattices and p and q are, respectively, their associated asymmetric norms (p(x) = ‖⁺‖, q(y) = ‖y ⁺‖), we prove that the positive operators from X to Y are elements of the cone LC(X, Y). We also study the dual space of an asymmetric normed space and finally we give open mapping and closed graph type theorems in the framework of asymmetric normed spaces. The classical results for normed spaces follow as particular cases. The author acknowledges the support of the Ministerio de Educación y Ciencia of Spain and FEDER, under grant MTM2006-14925-C02-01 and Generalitat Valenciana under grant GV/2007/198.     

Linear operators in finite dimensional probabilistic normed spaces

In this paper, we consider strongly bounded linear operators on a finite dimensional probabilistic normed space and define the topological isomorphism between probabilistic normed spaces. Then we prove that every finite dimensional probabilistic normed space which is a topological vector space is complete.     

The structure of weak Pareto solution sets in piecewise linear multiobjective optimization in normed spaces

In general normed spaces,we consider a multiobjective piecewise linear optimization problem with the ordering cone being convex and having a nonempty interior.We establish that the weak Pareto optimal solution set of such a problem is the union of finitely many polyhedra and that this set is also arcwise connected under the cone convexity assumption of the objective function.Moreover,we provide necessary and suffcient conditions about the existence of weak(sharp) Pareto solutions.     

Sufficient enlargements of minimal volume for finite-dimensional normed linear spaces

Let BY denote the unit ball of a normed linear space Y. A symmetric, bounded, closed, convex set A in a finite-dimensional normed linear space X is called a sufficient enlargement for X if, for an arbitrary isometric embedding of X into a Banach space Y, there exists a linear projection such that P(BY)⊂A. The main results of the paper: (1) Each minimal-volume sufficient enlargement is linearly equivalent to a zonotope spanned by multiples of columns of a totally unimodular matrix. (2) If a finite-dimensional normed linear space has a minimal-volume sufficient enlargement which is not a parallelepiped, then it contains a two-dimensional subspace whose unit ball is linearly equivalent to a regular hexagon.     

关于赋范空间上连续自映射的回归点

在赋范空间中讨论回归点的性质,主要得到了结果：（1）如果,是序列紧赋范空间X上的连续双射,x是f的任一回归点,则对于任意整数N〉0都存在f的回归点x0∈X使得f^n（x0）=x;（2）序列紧赋范空间上连续自映射的回归点集是f的强不变子集;（3）如果f是局部连通赋范空间X上的连续自映射,则f的每一个回归点或是类周期点或是类周期点的聚点．作为推论,在实直线段上得到了类似的结论．     

A characterization of normed spaces

We prove that the metric characterization of real normed spaces obtained by T. Oikhberg and H. Rosenthal can be obtained without a continuity assumption provided that the space is at least two-dimensional. In order to get this improvement we first need to understand the exceptional one-dimensional case.     

Some topologies on a Šerstnev probabilistic normed space

In this paper we will introduce two other topologies, coarser than the so-called strong topology, on a class of Šerstnev probabilistic normed spaces, and obtain some important properties of these topologies. We will show that under the first topology, denoted by τ₀, our probabilistic normed space is decomposable into the topological direct sum of a normable subspace and the subspace of probably null elements. Under the second topology, which is in fact the inductive limit topology of a family of locally convex topologies, the dual space becomes a locally convex topological vector space.     

The topological structure of a certain Menger space

In this paper we will reconsider the topological structure of Menger probabilistic normed spaces (briefly PN-spaces) under the t-norm M. We will prove that this topology is compatible with the topology induced by a countable and separating family of semi-norms, and hence the well-known theorems of classical functional analysis (such as the principle of uniform boundedness, open mapping and closed graph theorems) are valid in this context also. We will meanwhile obtain a method by which one may construct easily a large class of PN-spaces. Finally, using this method, we see that a certain subspace of bounded linear operators between PN-spaces, i.e. the class of strongly bounded linear operators, has a natural PN structure.     

Some remarks on functions with values in probabilistic normed spaces

In this paper we consider an enlargement of the notion of the probabilistic normed space. For this new class of probabilistic normed spaces we give some topological properties. By using properties of the probabilistic norm we prove some differential and integral properties of functions with values into probabilistic normed spaces. As special cases, results for deterministic and random functions can be obtained.      

The Goldstine Theorem for asymmetric normed linear spaces

It is well known that if (X,q) is an asymmetric normed linear space, then the function qs defined on X by qs(x)=max{q(x),q(−x)}, is a norm on the linear space X. However, the lack of symmetry in the definition of the asymmetric norm q yields an algebraic asymmetry in the dual space of (X,q). This fact establishes a significant difference with the standard results on duality that hold in the case of locally convex spaces. In this paper we study some aspects of a reflexivity theory in the setting of asymmetric normed linear spaces. In particular, we obtain a version of the Goldstine Theorem to these spaces which is applied to prove, among other results, a characterization of reflexive asymmetric normed linear spaces.     

关于Z-空间的对偶空间的讨论

本文以自然的方式定义了从Z-空间X到Z-空间Y的有界线性算子的和以及它们的数乘.从而得到了与赋范空间的对偶空间理论类似的一系列结论.     

The Chebyshev centers in a normed linear space

ＳＯＮＧＷＥＮＨＵＡ（宋文华）（ＩｎｓｔｉｔｕｔｅｏｆＭａｔｈｅｍａｔｉｃａｌＳｃｉｅｎｃｅ,ＤａｌｉａｎＵｎｉｖｅｒｓｉｔｙｏｆＴｅｃｈｎｏｌｏｇｙ,Ｄａｌｉａｎ１１６０２４,Ｃｈｉｎａ）Ａｂｓｔｒａｃｔ：ＡｃｈａｒａｃｔｅｒｉｚａｔｉｏｎｏｆＣｈ...     

On a class of real normed lattices

We say that a real normed lattice is quasi-Baire if the intersection of each sequence of monotonic open dense sets is dense. An example of a Baire-convex space, due to M. Valdivia, which is not quasi-Baire is given. We obtain that E is a quasi-Baire space iff is a pairwise Baire bitopological space, where , is a quasi-uniformity that determines, in L. Nachbin's sense, the topological ordered space E.     

Reduced bodies in Minkowski space

Summary We study the existence of almost-periodic solutions for a second order abstract Cauchy problem defined in a Banach space.     

等距线性延拓问题

本文引入了两个赋范空间的单位球面间等距映射的延拓问题,并且列出了相关问题的一些重要结果和近期的进展.     

D-boundedness andD-compactness in finite dimensional probabilistic normed spaces

In this paper, we prove that in a finite dimensional probabilistic normed space, every two probabilistic norms are equivalent and we study the notion ofD-compactness and D-boundedness in probabilistic normed spaces.     

On some properties of Lipschitz mappings of the real line into a normed space

We prove that for each normed space Y of infinite dimension and each porous set E ? ? there exists a Lipschitz mapping f: ? → Y such that the graph of f has a tangent at each of its points and f is not differentiable at any point of E. In this article we continue our research in [1] on contingents.     

Halving circular arcs in normed planes

Studying the relation between the length of a chord of the unit circle and the length of the arc corresponding to it, some new characterizations of the Euclidean plane among all normed planes are obtained. All these results yield characterizations of inner product spaces in higher dimensions. Supported by the National Natural Science Foundation of China, grant number 10671048.     

On isometric extension problem between two unit spheres

In this paper we introduce the isometric extension problem of isometric mappings between two unit spheres. Some important results of the related problems are outlined and the recent progress is mentioned.