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.

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.

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.

A Hake-Type Theorem for Integrals with Respect to Abstract Derivation Bases in the Riesz Space Setting

2015 ◽  
Vol 65 (6) ◽  
Author(s):  
A. Boccuto ◽  
V. A. Skvortsov ◽  
F. Tulone
Keyword(s):  
Measure Space ◽  
Type Theorem ◽  
Riesz Space ◽  
Riesz Spaces ◽  
Space Setting ◽  
Type Integral ◽  
Topological Measure ◽  
Technical Properties

AbstractA Kurzweil-Henstock type integral with respect to an abstract derivation basis in a topological measure space, for Riesz space-valued functions, is studied. A Hake-type theorem is proved for this integral, by using technical properties of Riesz spaces.

Riesz spaces with the order-continuity property. I

1977 ◽  
Vol 81 (1) ◽  
pp. 31-42 ◽  
Author(s):  
D. H. Fremlin
Keyword(s):  
Riesz Space ◽  
Continuity Property ◽  
Sequential Order ◽  
Order Continuity ◽  
Linear Map ◽  
Riesz Spaces ◽  
Quotient Spaces ◽  
Positive Linear Map ◽  
Order Continuous

A Riesz space E has the (sequential) order-continuity property if every positive linear map from E to an Archimedean Riesz space is (sequentially) order-continuous. This is the case if and only if the canonical maps from E to its Archimedean quotient spaces are all (sequentially) order-continuous. I relate these properties to others that have been described elsewhere.

Riesz spaces with the order-continuity property. II

1978 ◽  
Vol 83 (2) ◽  
pp. 211-223 ◽  
Author(s):  
D. H. Fremlin
Keyword(s):  
Internal Structure ◽  
Riesz Space ◽  
Continuity Property ◽  
Order Continuity ◽  
Linear Map ◽  
Riesz Spaces ◽  
Positive Linear Map ◽  
Order Continuous

I continue to investigate Riesz spaces E with the property that every positive linear map from E to an Archimedean Riesz space is sequentially order-continuous. In order to give a criterion for the product of such spaces to be another, we are forced to investigate their internal structure, and to develop an ordinal hierarchy of such spaces.

scholarly journals Ideal convergence in locally solid Riesz spaces

Filomat ◽  
2014 ◽  
Vol 28 (4) ◽  
pp. 797-809 ◽  
Author(s):  
Bipan Hazarika
Keyword(s):  
Bounded Sequence ◽  
Riesz Space ◽  
Ideal Convergence ◽  
Riesz Spaces ◽  
Basic Properties ◽  
Locally Solid Riesz Space ◽  
Positive Integers ◽  
The Ideal

An ideal I is a family of subsets of positive integers N which is closed under taking finite unions and subsets of its elements. In this paper, we introduce the concepts of ideal ?-convergence, ideal ?-Cauchy and ideal ?-bounded sequence in locally solid Riesz space endowed with the topology ?. Some basic properties of these concepts has been investigated. We also examine the ideal ?-continuity of a mapping defined on locally solid Riesz space.

scholarly journals On generalized statistical convergence and boundedness of Riesz space-valued sequences

Filomat ◽  
2019 ◽  
Vol 33 (15) ◽  
pp. 4989-5002
Author(s):  
Sudip Pal ◽  
Sagar Chakraborty
Keyword(s):  
Statistical Convergence ◽  
Necessary Conditions ◽  
Convergent Sequence ◽  
Bounded Sequence ◽  
Riesz Space ◽  
Riesz Spaces ◽  
Natural Density

We consider the notion of generalized density, namely, the natural density of weight 1 recently introduced in [4] and primarily study some sufficient and almost converse necessary conditions for the generalized statistically convergent sequence under which the subsequence is also generalized statistically convergent. Also we consider similar types of results for the case of generalized statistically bounded sequence. Some results are further obtained in a more general form by using the notion of ideals. The entire investigation is performed in the setting of Riesz spaces extending the recent results in [13].

Inextensible Riesz spaces

1975 ◽  
Vol 77 (1) ◽  
pp. 71-89 ◽  
Author(s):  
D. H. Fremlin
Keyword(s):  
Measurable Cardinal ◽  
Riesz Space ◽  
Abstract Characterization ◽  
Riesz Spaces ◽  
The Third ◽  
The Subject

My aim in this paper is to give an abstract characterization of the C∞ spaces described in (6) or (9), and to develop some of the remarkable special properties of these spaces. Although the subject is in some ways highly specialized, inextensible and sequentially inextensible spaces seem common enough (they include all spaces of the forms Rx and L0) to be worth studying, and I have already employed them in the proof of more general results (1).In the first section I set out those properties that can be described in simple Riesz space terms; much of this work has already been published in slightly different forms. In the second part I go on to questions that arise when we impose a topology on an inextensible Riesz space. Finally, in the third section, I discuss some problems, arising from the work before, which are related to the famous measurable cardinal problem.

Abstract Köthe spaces. IV

1968 ◽  
Vol 64 (1) ◽  
pp. 45-52
Author(s):  
D. H. Fremlin
Keyword(s):  
Tensor Product ◽  
Function Space ◽  
Function Spaces ◽  
Riesz Space ◽  
Natural Order ◽  
Order Structure ◽  
Topological Properties ◽  
Projective Tensor Product ◽  
Köthe Spaces ◽  
Projective Tensor

In this paper I investigate the completed projective tensor product of two perfect Riesz spaces, and show how a natural order structure on this renders it also a perfect Riesz space. Sections 7–14 contain interesting order-topological properties of this tensor product. Finally, section 15 describes how the tensor product of function spaces may be represented as a function space, in the manner of (1 b).

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