ON VECTOR SPACES WITH LOCALLY BOUNDED CONVERGENCE STRUCTURES

ABSTRACT

It is known that a precompact set in a limit vector space is not necessarily bounded. In this paper it is shown that the topological vector space result that every precompact set is bounded can be extended to the large class of locally bounded prelimit vector spaces. This result is used to extend the well-known characterization of finite-dimensional separated topological vector spaces to locally bounded prelimit vector spaces.     

Certain Norm Related Quantities of Unbounded Linear Operators

Certain norm related functions of linear operators are considered in the very general setting of not necessarily continuous linear operators in normed spaces. These are shown to be closely related to the theory of precompact, strictly cosingular and a class of Φ_? type operators in addition to having applications to perturbation theory. We also obtain some basic continuity and precompactness properties of linear operators in normed spaces which are expressed in terms of the functions under consideration.     

模糊赋范线性空间的紧性与完备性

讨论了Fuzzy赋范线性中准紧集、完备集及有界集间的关系;给出完备Fuzzy赋范空间的闭球套定理与Baire定理;刻画了了有限维Fuzzy赋范空间的特征。     

A note on precompact uniform frames

Precompactness or total boundedness for uniform frames is usually distinguished by a cover approach. In this note, we provide alternate characterizations of precompact uniform frames. In particular, we formulate pointfree filter analogues of various classical topological results on precompactness. We also revisit the notion of convergence and clustering of filters in a frame and introduce weakly Cauchy filters and strong Cauchy completeness in the setting of uniform frames.     

Lokalkreisförmige Vektorgruppen

We define locally circled vector groups as topological vector spaces over the discrete real or complex numberfield with a neighbourhoodbase of zero consisting of circled sets. Every topological vector space is a locally circled vector group. Topological vector groups and especially locally circled vector groups have useful applications to topological vector spaces and this paper is intended as an introduction to the theory of locally circled vector groups. Continuations including applications to topological vector spaces will follow. Here we study the structure of finite dimensional and locally compact vector groups, describe those locally circled vector groups which have a generating precompact circled set and finally prove some theorems about convex sets in these spaces.     

Compactness in Spaces of Vector Valued Continuous Functions and Asymptotic Almost Periodicity

We characterize the precompact sets in spaces of vector valued continuous functions and use the resulting criteria to investigate asymptotic behaviour of such functions defined on a halfline. This problem arose in the context of a qualitative study of solutions to the abstract Cauchy problem. We give particular consideration to the relationship between vector valued asymptotically almost periodic functions on a subinterval [α, ∞] of the real line and precompactness of the set of its translates. Our compactness criteria are also applied to a question concerning the approximation property for spaces of vector valued continuous functions with topologies induced by weighted analogues of the supremum norm. as well as to obtain nonlinear variants on factorization of compact operators through reflexive Banach spaces.     

PRECOMPACTNESS IN FUZZY UNIFORM SPACES

Abstract

The notion of a precompact fuzzy set in a fuzzy uniform space is defined and it is shown that this is a good extension of the standard notion. A theory of precompact fuzzy sets is developed using the previously defined notion of a Cauchy prefilter in a fuzzy uniform space and this theory generalises standard theory.     

Products of straight spaces

A metric space X is straight if for each finite cover of X by closed sets, and for each real valued function f on X, if f is uniformly continuous on each set of the cover, then f is uniformly continuous on the whole of X. A locally connected space is straight iff it is uniformly locally connected (ULC). It is easily seen that ULC spaces are stable under finite products. On the other hand the product of two straight spaces is not necessarily straight. We prove that the product X×Y of two metric spaces is straight if and only if both X and Y are straight and one of the following conditions holds:

(a)

both X and Y are precompact;

(b)

both X and Y are locally connected;

(c)

one of the spaces is both precompact and locally connected.

In particular, when X satisfies (c), the product X×Z is straight for every straight space Z.Finally, we characterize when infinite products of metric spaces are ULC and we completely solve the problem of straightness of infinite products of ULC spaces.     

Matricial quantum Gromov-Hausdorff distance

We develop a matricial version of Rieffel's Gromov-Hausdorff distance for compact quantum metric spaces within the setting of operator systems and unital C∗-algebras. Our approach yields a metric space of “isometric” unital complete order isomorphism classes of metrized operator systems which in many cases exhibits the same convergence properties as those in the quantum metric setting, as for example in Rieffel's approximation of the sphere by matrix algebras using Berezin quantization. Within the metric subspace of metrized unital C∗-algebras we establish the convergence of sequences which are Cauchy with respect to a larger Leibniz distance, and we also prove an analogue of the precompactness theorems of Gromov and Rieffel.     

Statistical convergence of double sequences in intuitionistic fuzzy normed spaces

Recently, the concept of intuitionistic fuzzy normed spaces was introduced by Saadati and Park [Saadati R, Park JH. Chaos, Solitons & Fractals 2006;27:331–44]. Karakus et al. [Karakus S, Demirci K, Duman O. Chaos, Solitons & Fractals 2008;35:763–69] have quite recently studied the notion of statistical convergence for single sequences in intuitionistic fuzzy normed spaces. In this paper, we study the concept of statistically convergent and statistically Cauchy double sequences in intuitionistic fuzzy normed spaces. Furthermore, we construct an example of a double sequence to show that in IFNS statistical convergence does not imply convergence and our method of convergence even for double sequences is stronger than the usual convergence in intuitionistic fuzzy normed space.     

Compactness theorems for abstract evolution problems

This paper deals with precompactness of trajectories of solutions of general nonlinear evolution problems in Banach spaces in the compact resolvent case. We carry out new criterions which extend or supplement for instance fundamental results of Pazy, Dafermos-Slemrod or Webb. These criterions have staightforward applications to Dynamical Systems and Control Theory.     

Group duality with the topology of precompact convergence

The Pontryagin-van Kampen (P-vK) duality, defined for topological Abelian groups, is given in terms of the compact-open topology. Polar reflexive spaces, introduced by Köthe, are those locally convex spaces satisfying duality when the dual space is equipped with the precompact-open topology. It is known that the additive groups of polar reflexive spaces satisfy P-vK duality. In this note we consider the duality of topological Abelian groups when the topology of the dual is the precompact-open topology. We characterize the precompact reflexive groups, i.e., topological groups satisfying the group duality defined in terms of the precompact-open topology. As a consequence, we obtain a new characterization of polar reflexive spaces. We also present an example of a space which satisfies P-vK duality and is not polar reflexive. Some of our results respond to questions appearing in the literature.     

Boundedly Compatible Webs and Strict Mackey Convergence

We look for characterizations of those locally convex spaces that satisfy the strict Mackey convergence condition within the context of spaces with webs. We will say that a locally convex space has a boundedly compatible web if it has a web of absolutely convex sets whose members behave like zero neighborhoods in a metrizable locally convex space. It will be shown that these locally convex spaces satisfy the strict Mackey convergence condition. One consequence of this result will be a characterization of boundedly retractive inductive limits. We will also prove that if E is locally complete and webbed, then the strict Mackey convergence condition is equivalent to E having a boundedly compatible web.     

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.     

A general duality result for precompact sets

In this paper we present an elementary proof of a general duality result for precompact sets which can be considered as a far-reaching generalization of a well-known result of Grothendieck on precompactness in dual systems. It is then shown that a number of known results can be deduced from it, amongst others a general form of the Arzela-Ascoli theorem and Grothendieck's duality theorem itself.     

Compact and precompact sets in asymmetric locally convex spaces

The aim of the present paper is to study precompactness and compactness within the framework of asymmetric locally convex spaces, defined and studied by the author in [S. Cobza?, Asymmetric locally convex spaces, Int. J. Math. Math. Sci. 2005 (16) (2005) 2585-2608]. The obtained results extend some results on compactness in asymmetric normed spaces proved by [L.M. García-Raffi, Compactness and finite dimension in asymmetric normed linear spaces, Topology Appl. 153 (2005) 844-853], and [C. Alegre, I. Ferrando, L.M. García-Raffi, E.A. Sánchez-Pérez, Compactness in asymmetric normed spaces, Topology Appl. 155 (6) (2008) 527-539].     

On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space

The concept of statistical convergence was introduced by Fast [H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951) 241–244] which was later on studied by many authors. In [J.A. Fridy, C. Orhan, Lacunary statistical convergence, Pacific J. Math. 160 (1993) 43–51], Fridy and Orhan introduced the idea of lacunary statistical convergence. Quite recently, the concept of statistical convergence of double sequences has been studied in intuitionistic fuzzy normed space by Mursaleen and Mohiuddine [M. Mursaleen, S.A. Mohiuddine, Statistical convergence of double sequences in intuitionistic fuzzy normed spaces, Chaos Solitons Fractals (2008), doi:10.1016/j.chaos.2008.09.018]. In this paper, we study lacunary statistical convergence in intuitionistic fuzzy normed space. We also introduce here a new concept, that is, statistical completeness and show that IFNS is statistically complete but not complete.     

Generalized Dini theorems for nets of functions on arbitrary sets

We characterize the uniform convergence of pointwise monotonic nets of bounded real functions defined on arbitrary sets, without any particular structure. The resulting condition trivially holds for the classical Dini theorem. Our vector-valued Dini-type theorem characterizes the uniform convergence of pointwise monotonic nets of functions with relatively compact range in Hausdorff topological ordered vector spaces. As a consequence, for such nets of continuous functions on a compact space, we get the equivalence between the pointwise and the uniform convergence. When the codomain is locally convex, we also get the equivalence between the uniform convergence and the weak-pointwise convergence; this also merges the Dini-Weston theorem on the convergence of monotonic nets from Hausdorff locally convex ordered spaces. Most of our results are free of any structural requirements on the common domain and put compactness in the right place: the range of the functions.     

Convolution on distribution spaces characterized by regularization

Locally convex convolutor spaces are studied which consist of those distributions that define a continuous convolution operator mapping from the space of test functions into a given locally convex lattice of measures. The convolutor spaces are endowed with the topology of uniform convergence on bounded sets. Their locally convex structure is characterized via regularization and function-valued seminorms under mild structural assumptions on the space of measures. Many recent generalizations of classical distribution spaces turn out to be special cases of the general convolutor spaces introduced here. Recent topological characterizations of convolutor spaces via regularization are extended and improved. A valuable property of the convolutor spaces in applications is that convolution of distributions inherits continuity properties from those of bilinear convolution mappings between the locally convex lattices of measures.