Lattice-valued continuous convergence is induced by a lattice-valued uniform convergence structure

2006 ◽  
Vol 157 (20) ◽  
pp. 2715-2724
Author(s):  
Gunther Jäger
Keyword(s):  
Uniform Convergence ◽  
Continuous Convergence ◽  
Convergence Structure

The Schwartz property and nuclearity of spaces of smooth and holomorphic functions in infinite dimensions

1990 ◽  
Vol 107 (2) ◽  
pp. 377-385
Author(s):  
Sten Bjon
Keyword(s):  
Uniform Convergence ◽  
Open Subset ◽  
Convex Space ◽  
Holomorphic Functions ◽  
Locally Convex Space ◽  
Infinite Dimensions ◽  
Continuous Convergence ◽  
Convergence Structure ◽  
Locally Convex ◽  
Local Uniform Convergence

In [8] it was shown that a locally convex space E is a Schwartz space if and only if the convergence algebras Hc(U) and He(U) of holomorphic functions on an open subset of E coincide, i.e. if and only if continuous convergence c (see [1]) and the associated equable convergence structure e (= local uniform convergence, see [2, 13]) coincide.

scholarly journals Order compatibility for Cauchy spaces and convergence spaces

1987 ◽  
Vol 10 (2) ◽  
pp. 209-216
Author(s):  
D. C. Kent ◽  
Reino Vainio
Keyword(s):  
Uniform Convergence ◽  
Weak Order ◽  
Convergence Spaces ◽  
Convergence Structure ◽  
Cauchy Structure ◽  
Cauchy Spaces

A Cauchy structure and a preorder on the same set are said to be compatible if both arise from the same quasi-uniform convergence structure onX. Howover, there are two natural ways to derive the former structures from the latter, leading to “strong” and “weak” notions of order compatibility for Cauchy spaces. These in turn lead to characterizations of strong and weak order compatibility for convergence spaces.

scholarly journals A counterexample to the density of the space generated by the composition operators

1977 ◽  
Vol 16 (1) ◽  
pp. 79-81
Author(s):  
Ronald Beattie
Keyword(s):  
Vector Space ◽  
Dual Space ◽  
Composition Operators ◽  
Operator Space ◽  
Natural Analogue ◽  
Convergence Space ◽  
Continuous Convergence ◽  
Continuous Mappings ◽  
Convergence Structure

It is known that, for an arbitrary convergence space X, the vector space generated by X is dense in LcCc (X) where both C(X) and its dual space carry the continuous convergence structure. In this note, a natural analogue formulated for the operator space L(Cc(X), Cc(X)) is considered, namely: is the vector space generated by the composition operators associated to the continuous mappings in C(X, X) dense in Lc (Cc (X), Cc (X)) ? This question is answered in the negative by a counterexample.

scholarly journals A note on convergence of nets of multifunctions

Filomat ◽  
2018 ◽  
Vol 32 (6) ◽  
pp. 2019-2028
Author(s):  
Marian Przemski
Keyword(s):  
Uniform Convergence ◽  
Continuous Convergence

The purpose of this paper is to investigate some new types of continuous convergence and quasi-uniform convergence of nets of multifunctions. The main three theorems describe a general types of interrelationships between forms of convergence, the continuity of the limits of nets of multifunctions and the continuity of members of such nets.

scholarly journals The order topology for function lattices and realcompactness

1981 ◽  
Vol 4 (2) ◽  
pp. 289-304 ◽  
Author(s):  
W. A. Feldman ◽  
J. F. Porter
Keyword(s):  
Uniform Convergence ◽  
Continuous Functions ◽  
Order Topology ◽  
Convergence Space ◽  
Vector Lattices ◽  
Convergence Structure ◽  
Relative Uniform Convergence ◽  
Compact Convergence

A latticeK(X,Y)of continuous functions on spaceXis associated to each compactificationYofX. It is shown forK(X,Y)that the order topology is the topology of compact convergence onXif and only ifXis realcompact inY. This result is used to provide a representation of a class of vector lattices with the order topology as lattices of continuous functions with the topology of compact convergence. This class includes everyC(X)and all countably universally complete function lattices with 1. It is shown that a choice ofK(X,Y)endowed with a natural convergence structure serves as the convergence space completion ofVwith the relative uniform convergence.

scholarly journals Regular completions of Cauchy spaces via function algebras

1974 ◽  
Vol 11 (1) ◽  
pp. 77-88 ◽  
Author(s):  
R.J. Gazik ◽  
D.C. Kent
Keyword(s):  
Function Algebra ◽  
Universal Property ◽  
Continuous Convergence ◽  
Function Algebras ◽  
Cauchy Space ◽  
Convergence Structure ◽  
Cauchy Spaces

A regular completion with the universal property is obtained for each member of a certain class of Cauchy spaces by embedding the Cauchy space in a complete function algebra with the continuous convergence structure.

On the Convergence Vector Space ℒ,(E, F) and its Dual Space

1978 ◽  
Vol 21 (3) ◽  
pp. 279-284 ◽  
Author(s):  
Ronald Beattie
Keyword(s):  
Vector Space ◽  
Normed Space ◽  
Dual Space ◽  
Continuous Mapping ◽  
Continuous Linear ◽  
Continuous Convergence ◽  
Convergence Structure ◽  
Locally Convex

Let E be a locally convex tvs, F a normed space and the space of continuous linear mappings from E into F In this paper, we investigate the continuous convergence structure (c-structure) on. denotes the resulting convergence vector space (cvs).The c-structure is by definition the coarsest cvs structure on making evaluation a continuous mapping.

Links between Probabilistic Convergence Groups Under Triangular Norms and Enriched Lattice-Valued Convergence Groups

2016 ◽  
Vol 12 (02) ◽  
pp. 53-76
Author(s):  
T. M. G. Ahsanullah ◽  
Fawzi Al-Thukair
Keyword(s):  
Uniform Convergence ◽  
Point Of View ◽  
Close Connection ◽  
Uniform Structure ◽  
Triangular Norms ◽  
Convergence Group ◽  
Convergence Structure ◽  
Basic Facts ◽  
New Perspective ◽  
Categorical Point

We propose here two types of probabilistic convergence groups under triangular norms; present some basic facts, and give some characterizations for both the cases. We look at the possible link from categorical point of view between each of the proposed type and enriched lattice-valued convergence group. We produce several natural examples on probabilistic convergence groups under triangular norms. We also present a notion of probabilistic uniform convergence structure in a new perspective, showing that every probabilistic convergence group is probabilistic uniformizable. Moreover, we prove that this probabilistic uniform structure maintains a close connection with a known enriched lattice-valued uniform convergence structure.

scholarly journals Complete Heyting algebra-valued convergence semigroups

Filomat ◽  
2018 ◽  
Vol 32 (2) ◽  
pp. 619-633
Author(s):  
T.M.G. Ahsanullah ◽  
Fawzi Al-Thukair
Keyword(s):  
Uniform Convergence ◽  
Heyting Algebra ◽  
Cancellative Semigroup ◽  
Convergence Space ◽  
Convergence Group ◽  
Convergence Structure ◽  
The Subject ◽  
Basic Facts

Considering a complete Heyting algebra H, we introduce a notion of stratified H-convergence semigroup. We develop some basic facts on the subject, besides obtaining conditions under which a stratified H-convergence semigroup is a stratified H-convergence group. We supply a variety of natural examples; and show that every stratified H-convergence semigroup with identity is a stratified H-quasiuniform convergence space. We also show that given a commutative cancellative semigroup equipped with a stratified H-quasi-unifom structure satisfying a certain property gives rise to a stratified H-convergence semigroup via a stratified H-quasi-uniform convergence structure.

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