The order-bound topology on Riesz spaces

1970 ◽  
Vol 67 (3) ◽  
pp. 587-593 ◽  
Author(s):  
Yau-Chuen Wong
Keyword(s):  
Convex Sets ◽  
Vector Lattice ◽  
Positive Cone ◽  
Riesz Space ◽  
Hausdorff Topology ◽  
Riesz Spaces ◽  
Neighbourhood Base ◽  
Solid Hull ◽  
Locally Convex

1. Introduction. Let (X, C) be a Riesz space (or vector lattice) with positive cone C. A subset B of X is said to be solid if it follows from |x| ≤ |b| with b in B that x is in B (where |x| denotes the supremum of x and − x). The solid hull of B (absolute envelope of B in the terminology of Roberts (2)) is denoted to be the smallest solid set containing B, and is denoted by SB. A locally convex Hausdorff topology on (X, C) is called a locally solid topology if admits a neighbourhood-base of 0 consisting of solid and convex sets in X; and (X, C, ), where is a locally solid topology, is called a locally convex Riesz space.

Abstract Köthe Spaces. I

1967 ◽  
Vol 63 (3) ◽  
pp. 653-660 ◽  
Author(s):  
D. H. Fremlin
Keyword(s):  
Riesz Space ◽  
Riesz Spaces ◽  
Köthe Spaces ◽  
Solid Hull

The purpose of this paper and the next is to demonstrate that the ‘perfect Riesz spaces’ of (1) are an effective abstraction of the ‘espaces de Köthe’ of (2). I shall follow the ideas of (1), with certain changes in notation:If L is a Riesz space and x, y ∈ L, let us denote sup (x, y) by x ∧ y and inf (x, y) by x ∧ y. I shall use the convenient if informal notation xr↓ ((1), section 16·1) and shall in this usage assume that 0 ∈ {r} and that x0 ≥ xτ for all τ. A set A ⊆ L is solid if x ∈ A and |y| ≤ |x| together imply that y ∈ A; A is then balanced. The solid hull of A is the set {y: ∃ x ∈ A, |y| ≤ |x|}; this is the smallest solid set containing A. An ‘ideal’ ((1), section 17) is then a solid subspace.

An unpleasant set in a non-locally-convex vector lattice

1973 ◽  
Vol 18 (3) ◽  
pp. 229-233
Author(s):  
J. D. Pryce
Keyword(s):  
Convex Hull ◽  
Vector Lattice ◽  
Linear Topological Space ◽  
Real Vector ◽  
Vector Lattices ◽  
Köthe Spaces ◽  
The Common ◽  
Solid Hull ◽  
Image Position ◽  
Locally Convex

In a linear topological space E one often carries out various “ smoothing ” operations on a subset A, such as taking the convex hull co A and the closure A-. If E is also a (real) vector lattice, the solid hullis also a natural “ smoothing out ” of A. If sol A = A then A is called solid, and if E has a base of solid neighbourhoods of 0 as do all the common topological vector lattices such as C(X), Lp, Köthe spaces and so on—then E is called a locally solid space.

scholarly journals A Study on Fuzzy Order Bounded Linear Operators in Fuzzy Riesz Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1512
Author(s):  
Juan Luis García Guirao ◽  
Mobashir Iqbal ◽  
Zia Bashir ◽  
Tabasam Rashid
Keyword(s):  
Riesz Space ◽  
Linear Operators ◽  
Separation Property ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Riesz Spaces ◽  
Fuzzy Order ◽  
Locally Convex ◽  
Special Case

This paper aims to study fuzzy order bounded linear operators between two fuzzy Riesz spaces. Two lattice operations are defined to make the set of all bounded linear operators as a fuzzy Riesz space when the codomain is fuzzy Dedekind complete. As a special case, separation property in fuzzy order dual is studied. Furthermore, we studied fuzzy norms compatible with fuzzy ordering (fuzzy norm Riesz space) and discussed the relation between the fuzzy order dual and topological dual of a locally convex solid fuzzy Riesz space.

Small Riesz spaces

1989 ◽  
Vol 105 (3) ◽  
pp. 523-536 ◽  
Author(s):  
G. Buskes ◽  
A. van Rooij
Keyword(s):  
Representation Theory ◽  
Direct Method ◽  
Riesz Space ◽  
Pictorial Representation ◽  
Representation Theorems ◽  
Riesz Spaces ◽  
Number Of Authors

Many facts in the theory of general Riesz spaces are easily verified by thinking in terms of spaces of functions. A proof via this insight is said to use representation theory. In recent years a growing number of authors has successfully been trying to bypass representation theorems, judging them to be extraneous. (See, for instance, [9,10].) In spite of the positive aspects of these efforts the following can be said. Firstly, avoiding representation theory does not always make the facts transparent. Reading the more cumbersome constructions and procedures inside the Riesz space itself one feels the need for a pictorial representation with functions, and one suspects the author himself of secret heretical thoughts. Secondly, the direct method leads to repeating constructions of the same nature over and over again.

scholarly journals A Fubini Theorem in Riesz spaces for the Kurzweil-Henstock Integral

2011 ◽  
Vol 9 (3) ◽  
pp. 283-304 ◽  
Author(s):  
A. Boccuto ◽  
D. Candeloro ◽  
A. R. Sambucini
Keyword(s):  
Type Theorem ◽  
Riesz Space ◽  
Fubini Theorem ◽  
Real Plane ◽  
Henstock Integral ◽  
Riesz Spaces

A Fubini-type theorem is proved, for the Kurzweil-Henstock integral of Riesz-space-valued functions defined on (not necessarily bounded) subrectangles of the “extended” real plane.

scholarly journals Nonsmooth analysis and optimization on partially ordered vector spaces

1992 ◽  
Vol 15 (1) ◽  
pp. 65-81 ◽  
Author(s):  
Thomas W. Reiland
Keyword(s):  
Vector Lattice ◽  
Vector Spaces ◽  
Range Space ◽  
Topological Vector Spaces ◽  
Complete Vector Lattice ◽  
Lipschitz Mappings ◽  
Ordered Vector Spaces ◽  
Partially Ordered Vector Spaces ◽  
Complete Vector ◽  
Locally Convex

Interval-Lipschitz mappings between topological vector spaces are defined and compared with other Lipschitz-type operators. A theory of generalized gradients is presented when both spaces are locally convex and the range space is an order complete vector lattice. Sample applications to the theory of nonsmooth optimization are given.

An n + 1 Member Decomposition for Sets Whose Lnc Points Form n Convex Sets

1975 ◽  
Vol 27 (6) ◽  
pp. 1378-1383 ◽  
Author(s):  
Marilyn Breen
Keyword(s):  
Convex Sets ◽  
Local Convexity ◽  
Locally Convex

Let S be a subset of Rd. A point x in 5 is a point of local convexity of S if and only if there is some neighborhood N of x such that, if y, z ∈ N ᑎ 5, then [y, z] ⊆ S. If S fails to be locally convex at some point q in S then q is called a point of local nonconvexity (lnc point) of S.

Some Characterizations of The Projection Property in Archimedean Riesz Spaces

1972 ◽  
Vol 24 (2) ◽  
pp. 306-311 ◽  
Author(s):  
K. K. Kutty ◽  
J. Quinn
Keyword(s):  
Band Structure ◽  
Prime Ideal ◽  
Vector Lattice ◽  
Strong Relationship ◽  
Ideal Structure ◽  
Projection Property ◽  
Dedekind Completion ◽  
Riesz Spaces

In this paper we give some new characterizations of the projection property in Archimedean Riesz spaces. Our approach primarily explores the interrelationships between such things as the band structure or the prime ideal structure of an Archimedean vector lattice and corresponding structures of its Dedekind completion. Our results show that, in general, there is a ‘strong“ relationship if and only if the original vector lattice has the projection property. The main result of this paper is Theorem 2.6 which both summarizes and extends all of the results we obtain prior to it.

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