Characterization of inner product spaces in V- and L-spaces

In [4], Freese and Murphy introduce a new class of spaces, the V-spaces, which include Banach spaces, hyperbolic spaces, and other metric spaces. In this class of spaces they investigate conditions which are equivalent to strict convexity in Banach spaces, and extend some of the Banach space results to this new class of spaces. It is natural to ask if known characterizations of real inner product spaces among Banach spaces can also be extended to this larger class of spaces. In the present paper it will be shown that a metrization of an angle bisector property used in [3] to characterize real inner product spaces among Banach spaces also characterizes real inner product spaces among V-spaces, and among another class of spaces, the L-spaces, which include hyperbolic spaces and strictly convex Banach spaces. In the process it is shown that in a complete, convex, externally convex metric space M, if the foot of a point on a metric line is unique, then M satisfies the monotone property, thus answering a question raised in [4].     

Fuzzy φψ-continuous Functions between L-topological Spaces (Ⅱ)

In the present paper, sum operation on sum spaces of a family of L-topological spaces is defined. Fuzzy φψ-continuity from sum spaces of a family of L-topological spaces into L-product spaces and from sum spaces of a family of L-topological spaces into L-topological spaces are investigated.     

Uniformity Structures on L-Fuzzy Spaces

We extend the notion of a uniform space in a natural way by defining a uniform spaces in L-fuzzy spaces.Although these spaces seem quite similar to ordinary case,we show that the category of this uniform spaces is a good extension of the category of ordinary uniform spaces and the category of L-uniform spaces.Moreover,we introduce the concept of uniform topological spaces in the framework of uniform spaces in L-fuzzy spaces.Furthermore,the relation between proximity and uniform spaces in L-fuzzy spaces will...     

Local connectedness in merotopic spaces

Abstract

In this paper uniformly locally uniformly connected merotopic spaces are studied. It turns out that their structural behaviour is essentially similar to that one of locally connected topological spaces. The introduced concept is also investigated for spaces of functions between filter-merotopic spaces (e.g. topological spaces, proximity spaces, convergence spaces) and the relationship to other concepts of local connectedness is clarified. In particular, the category of uniformly locally uniformly connected filter-merotopic spaces is Cartesian closed.     

Lorentz序列空间的装球问题

Ｂａｎａｃｈ空间中装球问题的研究，近四十年来已取得了令人瞩目的发展。Ｂａｎａｃｈ空间的装球值的范围已经确定，Ｌ＿ｐ空间及Ｏｒｌｉｃｚ序列空间Ｉ＿Ｍ等许多经典Ｂａｎａｃｈ空间装球值已经找到．本文研究又一类经典Ｂａｎａｃｈ空间──Ｌｏｒｅｎｔｚ序列空间的装球问题，给出了Ｌｏｒｅｎｔｚ序列空间的装球值。     

Straight nearness spaces

Straight spaces are spaces for which a continuous map defined on the space which is uniformly continuous on each set of a finite closed cover is then uniformly continuous on the whole space. Previously, straight spaces have been studied in the setting of metric spaces. In this paper, we present a study of straight spaces in the more general setting of nearness spaces. In a subcategory of nearness spaces somewhat more general than uniform spaces, we relate straightness to uniform local connectedness. We investigate category theoretic situations involving straight spaces. We prove that straightness is preserved by final sinks, in particular by sums and by quotients, and also by completions.     

LOCAL COMPACTNESS IN SEMI-UNIFORM CONVERGENCE SPACES

Abstract

Local compactness is studied in the highly convenient setting of semi-uniform convergence spaces which form a common generalization of (symmetric) limit spaces (and thus of symmetric topological spaces) as well as of uniform limit spaces (and thus of uniform spaces). It turns out that it leads to a cartesian closed topological category and, in contrast to the situation for topological spaces, the local compact spaces are exactly the compactly generated spaces. Furthermore, a one-point Hausdorff compactification for noncompact locally compact Hausdorff convergence spaces is considered.¹     

On Function Spaces with Mixed Norms — A Survey

The targets of this article are threefold. The first one is to give a survey on the recent developments of function spaces with mixed norms, including mixed Lebesgue spaces, iterated weak Lebesgue spaces, weak mixed-norm Lebesgue spaces and mixed Morrey spaces as well as anisotropic mixed-norm Hardy spaces. The second one is to provide a detailed proof for a useful inequality about mixed Lebesgue norms and the Hardy–Littlewood maximal operator and also to improve some known results on the maximal function characterizations of anisotropic mixed-norm Hardy spaces and the boundedness of Calderón–Zygmund operators from these anisotropic mixed-norm Hardy spaces to themselves or to mixed Lebesgue spaces. The last one is to correct some errors and seal some gaps existing in the known articles.     

关于sn-度量空间

本文利用ｓｎ－度量空间的一些等价刻划及ｓｎ－度量空间与ｇ－度量空间、度量空间之间的关系，研究了ｓｎ－度量空间的一些映射性质，证明了ｓｎ－度量空间的闭象是ｓｎ－度量空间当且仅当它是ｓｎ－第一可数的，利用这一结果证明了有限到一闭映射和开闭映射均保持ｓｎ－度量空间，并给出反例说明完备映射不保持ｓｎ－度量空间，本文还证明了ｓｎ－变量空问满足完备逆象Ｇδ－对角线定理．     

Compact adjoint operators and approximation properties

This paper is concerned with the space of all compact adjoint operators from dual spaces of Banach spaces into dual spaces of Banach spaces and approximation properties. For some topology on the space of all bounded linear operators from separable dual spaces of Banach spaces into dual spaces of Banach spaces, it is shown that if a bounded linear operator is approximated by a net of compact adjoint operators, then the operator can be approximated by a sequence of compact adjoint operators whose operator norms are less than or equal to the operator norm of the operator. Also we obtain applications of the theory and, in particular, apply the theory to approximation properties.     

广义Calderón-Zygmund算子交换子在Hardy型空间上的有界性

[b,T]表示由Lipschitz函数b与广义Calderon-Zygmund算子T生成的交换子．本文研究了[b,T]在经典Hardy空间和Herz型Hardy空间上的有界性,并且在临界点情形证明了该交换子是从Hardy空间到弱Lebesgue空间以及Herz型Hardy到弱Herz空间有界的．     

Weak Morrey spaces with applications

The main object of this investigation is to study weak Morrey spaces. Block spaces, which are preduals of weak Morrey spaces, are characterized. Besides, the Fatou property of block spaces is proved. Finally, as an application, we study the boundedness of singular integral operators in weak Morrey spaces.     

Dyadic Nikol’skii-Besov spaces and their relationship to classical spaces

Properties of dyadic spaces are considered and a relationship between dyadic and classical spaces is described. A property of classical spaces is proved by using the technique of dyadic spaces.     

Properties of ultraproducts

A direct proof is given of the completeness of ultraproduct spaces when the factor spaces are Banach spaces. The method of proof also shows the completeness of the ultraproduct even when the factor spaces are not complete for many (possibly all) ultrafilters. It is also shown that ultraproduct spaces are separable only when they are finite dimensional.     

Uniform universal covers of uniform spaces

We develop a generalized covering space theory for a class of uniform spaces called coverable spaces. Coverable spaces include all geodesic metric spaces, connected and locally pathwise connected compact topological spaces, in particular Peano continua, as well as more pathological spaces like the topologist's sine curve. The uniform universal cover of a coverable space is a kind of generalized cover with universal and lifting properties in the category of uniform spaces and uniformly continuous mappings. Associated with the uniform universal cover is a functorial uniform space invariant called the deck group, which is related to the classical fundamental group by a natural homomorphism. We obtain some specific results for one-dimensional spaces.     

Locale的弱拓扑表达

本文引入了弱拓扑空间的概念,证明了locale范畴与弱拓扑空间范畴的关系类似于拓扑空间范畴与locale范畴的关系。locale范畴严格包含于弱拓扑空间范畴并且与Sober的弱拓扑空间范畴等价。     

Hardy-Hausdorff spaces on the Heisenberg group

In this paper, by using the tent spaces on the Siegel upper half space, which are defined in terms of Choquet integrals with respect to Hausdorff capacity on the Heisenberg group, the Hardy-Hausdorff spaces on the Heisenberg group are introduced. Then, by applying the properties of the tent spaces on the Siegel upper half space and the Sobolev type spaces on the Heisenberg group, the atomic decomposition of the Hardy-Hausdorff spaces is obtained. Finally, we prove that the predual spaces of Q spaces on the Heisenberg group are the Hardy-Hausdorff spaces.     

PROXIMAL LIMIT SPACES

Abstract

The proximal limit spaces are introduced which fill the gap arising from the existence of proximity spaces, uniform spaces, and uniform limit spaces. It is shown that the proximal limit spaces can be considered as a bireflective subcategory of the topological category of uniform limit spaces. A limit space is induced by a proximal limit space if and only if it is a S₁-limit space.     

k半层空间的性质及相关问题

k半层空间是层空间与N空间的共同推广,本文综述k半层空间的研究进展,介绍k半层空间的刻画、性质及其推广,给出一些相关问题,供进一步研究.     

Interpolation of Bilinear Operators Between Quasi-Banach Spaces

We study interpolation, generated by an abstract method of means, of bilinear operators between quasi-Banach spaces. It is shown that under suitable conditions on the type of these spaces and the boundedness of the classical convolution operator between the corresponding quasi-Banach sequence spaces, bilinear interpolation is possible. Applications to the classical real method spaces, Calderón-Lozanovsky spaces, and Lorentz-Zygmund spaces are presented. The author is supported by the National Science Foundation under grant DMS 0099881. The author is supported by KBN Grant 1 P03A 013 26.