DOCSLIB.ORG

Laplace Transformation on Ordered Linear Space of Generalized Functions K

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Laplace Transformation on Ordered Linear Space of Generalized Functions K World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:2, No:3, 2008 Laplace Transformation on Ordered Linear Space of Generalized Functions K. V. Geetha and N. R. Mangalambal Abstract—Aim. We have introduced the notion of order to multi- Let La,b denote the linear space of all complex valued normed spaces and countable union spaces and their duals. The smooth functions defined on R. Let topology of bounded convergence is assigned to the dual spaces. The aim of this paper is to develop the theory of ordered topological linear L (t)=eat, 0 ≤ t<∞ a,b spaces La,b, L (w, z), the dual spaces of ordered multinormed spaces = ebt, −∞ <t<0. La,b, ordered countable union spaces L(w, z), with the topology of bounded convergence assigned to the dual spaces. We apply Laplace (Km) be a sequence of compact subsets of R such that K1 ⊆ transformation to the ordered linear space of Laplace transformable ⊆ R generalized functions. We ultimately aim at finding solutions to non- K2 ... and such that each compact subset of is contained homogeneous nth order linear differential equations with constant in one Km. Define coefficients in terms of generalized functions and comparing different ( ) = sup | ( ) k ( )| =0 1 2 solutions evolved out of different initial conditions. γKm,k φ La,b t D φ t ,k , , ... ∈ Method. The above aim is achieved by t Km { }∞ L L • Defining the spaces La,b, L(w, z). γKm,k k=0 is a multinorm on a,b,Km where a,b,Km is • Assigning an order relation on these spaces by identifying a the subspace of La,b whose elements have their support in positive cone on them and studying the properties of the cone. K . The above multinorm generates the topology τ on L L(w, z) m a,b,Km • Defining an order relation on the dual spaces a,b, L L L L(w, z) a,b,Km . a,b is equipped with the inductive limit topology of a,b, and assigning a topology to these dual spaces L which makes the order dual and the topological dual the same. τa,b as Km varies over all compact sets K1,K2,.... a,b is −st • Defining the adjoint of a continuous map on these spaces complete for τa,b. For each fixed s, e ∈La,b if and only k −st and studying its behaviour when the topology of bounded if a ≤ Re s ≤ b. For each positive integer k, t e ∈La,b if convergence is assigned to the dual spaces. and only if a<Re s<b. • Applying the two-sided Laplace Transformation on the ordered We recall the notions of a positive cone, normal cone and linear space of generalized functions W and studying some properties of the transformation which are used in solving strict b-cone that have been defined in [1]. differential equations. Definition 1: Let V be a multinormed space whose field of Result. The above techniques are applied to solve non-homogeneous scalars is R. A subset C or C(V ) is a positive cone in V if n-th order linear differential equations with constant coefficients in (i) C + C ⊆ C terms of generalized functions and to compare different solutions of (ii) αC ⊆ C for all scalars α>0 the differential equation. (iii) C ∩ (−C)={0} Keywords —Laplace transformable generalized function, positive When the field of scalars is C, C + iC is the positive cone in cone, topology of bounded convergence. V which is also denoted as C. C defines an order relation on V , φ ≤ ψ if ψ − φ ∈ C. Definition 2: Let V (τ) be an ordered multinormed space I. THE SPACES La,b, L(w, z) AND THEIR DUALS with positive cone C. C is normal for the topology τ generated We have associated the notion of ‘order’ to multinormed by the multinorm S if there is a neighbourhood basis of 0 for spaces, countable union spaces (see [3], for details of multi- τ consisting of full sets. normed spaces, countable union spaces) and their duals, by Definition 3: Let G be a saturated class of τ-bounded identifying a positive cone on these spaces. Also, the topology subsets of an ordered multinormed space V (τ) such that of bounded convergence is assigned to the dual spaces so that V = ∪{s : s ∈G}. The positive cone C in V (τ) is a strict the order dual and the topological dual become identical [1]. G-cone if the class G = {(S ∩ C) − (S ∩ C):S ∈G}is a International Science Index, Mathematical and Computational Sciences Vol:2, No:3, 2008 waset.org/Publication/13650 L L( ) In this section we define the spaces a,b, w, z and apply fundamental system for G. A strict G-cone for the the class G the above ideas to these spaces and to their duals. of all τ-bounded sets in V (τ) is called a strict b-cone. Definition 4: The positive cone C of La,b when La,b is Manuscript received March 1, 2008. restricted to real-valued functions is the set of all non-negative K. V. Geetha is with the Department of Mathematics, St. Joseph’s L + L College, Irinjalakuda, Kerala, India. Pin.680 121 (Phone: Off. 0480- functions in a,b. Then C iC is the positive cone in a,b 2825358, Res. 0480-2824613, cell. 9447994135, fax. 0480-2830954, email. which is also denoted as C. [email protected]) Now we prove that the cone of L is not normal but is a N. R. Mangalambal is with the Department of Mathematics, St. Joseph’s a,b College, Irinjalakuda, Kerala, India. Pin.680 121 (Phone: Off. 0480-2825358, strict b-cone. Res. 0480-2709858, cell. 9495246832, fax. 0480-2830954, email. thot- Theorem 1: The cone C of La,b is not normal. [email protected]) Proof: Let La,b be restricted to real-valued functions. Let m be a fixed positive integer and (φi) be a sequence of International Scholarly and Scientific Research & Innovation 2(3) 2008 200 scholar.waset.org/1307-6892/13650 World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:2, No:3, 2008 L ∩ = sup{ ( ): ∈ } functions in a,b,Km C such that λi φi t t Km Proof: Every positive linear functional is continuous for ( ) L L converges to 0 but φi does not converge to 0 for a,b,Km . the Schwartz topology τa,b on a,b. Define L+ = (L R) − (L R) a,b C a,b, C a,b, ψi = λi,t∈ Km+1 where C(La,b, R) is the linear subspace consisting of all non- =0,t∈ K +1. m negative order bounded linear functionals of L(La,b, R), the L Let ξ be the regularization of ψ defined by linear space of all order bounded, linear functionals on a,b. i i L+ ⊆L It follows that a,b a,b. L ( )= ( − ) ( ) On the other hand, the space a,b equipped with the ξi t θα t t ψi t dt . R (L L ) topology of bounded convergence β a,b, a,b and ordered ∈L ∀ C C L Then ξi a,b,Km+2 , i. by the dual cone of the cone in a,b is a reflexive space Also, 0 ≤ φi ≤ ξi, ∀i and (ξi) converges to 0 for τa,b.In ordered by a closed normal cone. Hence if D is a directed Proposition 1.3, Chapter 1, [2], it has been proved that the set (≤) of generalized functions that is either majorized in L (L L ) sup positive cone in an ordered linear space E(τ) is normal if and a,b or contains a β a,b, a,b -bounded section, then D L F( ) only if for any two nets {xβ : β ∈ I} and {yβ : β ∈ I} exists in a,b and the filter D of sections of D converges to sup (L L ) L+ = L in E(τ) if 0 ≤ xβ ≤ yβ for all β ∈ I and if {yβ : β ∈ I} D for β a,b, a,b . Hence a,b a,b and we conclude L (L L ) converges to 0 for τ then {xβ : β ∈ I} converges to 0 for τ.So that a,b with respect to the topology β a,b, a,b is both we conclude that C is not normal for the Schwartz topology. order complete and topologically complete. (See Proposition It follows that C + iC is not normal for La,b, we conclude 1.8, Chapter 4, [2]). ≤ ≤ ∈ that the cone of La,b is not normal. Theorem 4: Let a c, d b. The restriction of any f L L L L L Theorem 2: The cone C is a strict b-cone in La,b. a,b to c,d is c,d when a,b, c,d are assigned the topology Proof: Assume that La,b is restricted to real-valued of bounded convergence. functions. Let B be the saturated class of all bounded cir- Proof: When a ≤ c, d ≤ b, Lc,d ⊆La,b. The topology L L cled subsets of La,b for τa,b. Then La,b = ∪B∈BBc where τc,d of c,d is stronger than the topology induced on c,d by L ∈L Bc = {(B ∩ C) − (B ∩ C):B ∈ B} is a fundamental system a,b. By Zemanian [3], the restriction of any f a,b to L L L L for B and C is a strict b-cone since B is the class of all τ- c,d is in c,d when c,d and a,b are assigned the topology bounded sets in La,b.
Load more

Recommended publications
  • A Note on Riesz Spaces with Property-$ B$
    Czechoslovak Mathematical Journal Ş. Alpay; B. Altin; C. Tonyali A note on Riesz spaces with property-b Czechoslovak Mathematical Journal, Vol. 56 (2006), No. 2, 765–772 Persistent URL: http://dml.cz/dmlcz/128103 Terms of use: © Institute of Mathematics AS CR, 2006 Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz Czechoslovak Mathematical Journal, 56 (131) (2006), 765–772 A NOTE ON RIESZ SPACES WITH PROPERTY-b S¸. Alpay, B. Altin and C. Tonyali, Ankara (Received February 6, 2004) Abstract. We study an order boundedness property in Riesz spaces and investigate Riesz spaces and Banach lattices enjoying this property. Keywords: Riesz spaces, Banach lattices, b-property MSC 2000 : 46B42, 46B28 1. Introduction and preliminaries All Riesz spaces considered in this note have separating order duals. Therefore we will not distinguish between a Riesz space E and its image in the order bidual E∼∼. In all undefined terminology concerning Riesz spaces we will adhere to [3]. The notions of a Riesz space with property-b and b-order boundedness of operators between Riesz spaces were introduced in [1]. Definition. Let E be a Riesz space. A set A E is called b-order bounded in ⊂ E if it is order bounded in E∼∼. A Riesz space E is said to have property-b if each subset A E which is order bounded in E∼∼ remains order bounded in E.
    [Show full text]
  • Contents 1. Introduction 1 2. Cones in Vector Spaces 2 2.1. Ordered Vector Spaces 2 2.2
    ORDERED VECTOR SPACES AND ELEMENTS OF CHOQUET THEORY (A COMPENDIUM) S. COBZAS¸ Contents 1. Introduction 1 2. Cones in vector spaces 2 2.1. Ordered vector spaces 2 2.2. Ordered topological vector spaces (TVS) 7 2.3. Normal cones in TVS and in LCS 7 2.4. Normal cones in normed spaces 9 2.5. Dual pairs 9 2.6. Bases for cones 10 3. Linear operators on ordered vector spaces 11 3.1. Classes of linear operators 11 3.2. Extensions of positive operators 13 3.3. The case of linear functionals 14 3.4. Order units and the continuity of linear functionals 15 3.5. Locally order bounded TVS 15 4. Extremal structure of convex sets and elements of Choquet theory 16 4.1. Faces and extremal vectors 16 4.2. Extreme points, extreme rays and Krein-Milman's Theorem 16 4.3. Regular Borel measures and Riesz' Representation Theorem 17 4.4. Radon measures 19 4.5. Elements of Choquet theory 19 4.6. Maximal measures 21 4.7. Simplexes and uniqueness of representing measures 23 References 24 1. Introduction The aim of these notes is to present a compilation of some basic results on ordered vector spaces and positive operators and functionals acting on them. A short presentation of Choquet theory is also included. They grew up from a talk I delivered at the Seminar on Analysis and Optimization. The presentation follows mainly the books [3], [9], [19], [22], [25], and [11], [23] for the Choquet theory. Note that the first two chapters of [9] contains a thorough introduction (with full proofs) to some basics results on ordered vector spaces.
    [Show full text]
  • RIESZ SPACES " Given by Prof
    EUR 3140.Θ ÄO!·'· 1 \n\< tl '*·»ΤΗΗΊ)ίβ§"ίί IJM llHi 'li»**. 'tí fe É!>?jpfc EUROPEAN ATOMIC ENERGY COMMUNITY EURATOM LECTURES ON " RIESZ SPACES " given by Prof. A. C. ZAANEN iiil^{TT"1! i ■ BIT' '.'I . * . Ι'βΗΤ Editor : R. F. GLODEN !!i>M!?ÄÉ«il Siffifi 1966 loint Nuclear Research Center Ispra Establishment - Italy Scientific Information Processing Center - CETIS »»cm*if'-WW;flW4)apro 'BfiW»¡kHh;i.u·^: 2Λ*.;;, tf! :1Γ:«Μ$ ■ ■ J'ÎHO *sUr! if nb-. I;·"'-ii ;Γ^*Ε»"^Β1 hiik*MW!?5?'J.-i,K^ fill"; UP» LEGAL NOTICE This document was prepared under the sponsorship of the Commission of the European Atomic Energy Community (EURATOM). Neither the EURATOM Commission, its contractors nor any person acting- on their behalf : Make any warranty or representation, express or implied, with respect to the accuracy, completeness, or usefulness of the information contained in this document, or that the use of any information, apparatus, method, or process disclosed in this document may not infringe privately owned rights ; or Assume any liability with respect to the use of, or for damages resulting from the use of any information, apparatus, method or process disclosed in this document. This report is on sale at the addresses listed on cover page 4 at the price of FF 8.50 FB 85 DM 6.80 Lit. 1060 Fl. 6.20 When ordering, please quote the EUR number and the titl which are indicated on the cover of each report. m ■pejs!« EUR 3140.e LECTURES ON « RIESZ SPACES » given by Prof. A.C.
    [Show full text]
  • CALIFORNIA INSTITUTE of TECHNOLOGY L /D1
    DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA. CALIFORNIA 91125 EQUILIBRIA IN MARKETS WITH A RIESZ SPACE OF COMMODITIES Charalambos D. Aliprantis California Institute of Technology Indiana University and Purdue University at Indianapolis and �c,1\lUTEOF Donald J. Brown California Institute of Technology � 1:, � Yale University _,, � t:::0 :5 �,.� ,.ct'� � l� /d1�\ i /'l\ \ lf1: ...� � Slf�LL IA"'�\. SOCIAL SCIENCE WORKING PAPER 427 June 1982 ABSTRACT Using the theory of Riesz spaces, we present a new proof of the existence of competitive equilibria for an economy having a Riesz space of commodities. 2 EQUILIBRIA IN .MARKETS WITH A RIESZ SPACE OF COMMODITIES production. In proving existence, n�wley considers Lm(µ) with the sup norm topology, and in this case the dual space is the vector space of 1 • INTRODUCTION all bounded additive functionals on Lm(µ). n In the Arrow-Debreu model of a Walrasian economy, [3] and [7], The spaces 1R and Lm(µ) in addition to being ordered linear the commodity space is :mn and the price space is :m:. where n is the vector spaces are also Riesz spaces or vector lattices. In fact, number of commodities. Agent's characteristics such as consumption considered as Banach spaces under the sup norm, they belong to the sets, production sets, utility functions, the price simplex, excess special class of Banach lattices, i. e. to the class of normed Riesz n demand functions, etc. are introduced in terms of subsets of 1R or spaces which are complete under their norms. In this paper, we 1R: or functions on 1Rn or 1R:.
    [Show full text]
  • Riesz Spaces Valued Measures and Processes Bulletin De La S
    BULLETIN DE LA S. M. F. NASSIF GHOUSSOUB Riesz spaces valued measures and processes Bulletin de la S. M. F., tome 110 (1982), p. 233-257 <http://www.numdam.org/item?id=BSMF_1982__110__D233_0> © Bulletin de la S. M. F., 1982, tous droits réservés. L’accès aux archives de la revue « Bulletin de la S. M. F. » (http: //smf.emath.fr/Publications/Bulletin/Presentation.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/ conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Bull. 5<?c, ^.2ik. pr^ce, 110, 1982, p. 233-257. RIESZ SPACES VALUED MEASURES AND PROCESSES BY NASSIF GHOUSSOUB (*) ABSTRACT. — We give necessary and sufficient conditions for the weak convergence (resp. strong convergence, resp. order convergence) of L1-bounded (resp. uniformly bounded, resp. order bounded) supermartingales and, more generally, order asymptotic martingales valued in Banach lattices. RESUME. - On donne des conditions nccessaircs et suffisantes pour la convergence faible (resp. tone, resp. pour Fordre) des surmartingales et plus generalement des martingales asymptotiques pour Fordro a valeurs dans un treillis de Banach et qui sont bomees dans L1 (resp. uniformement bomees, resp. bomees pour 1'ordre). 0. Introduction This paper is mainly concerned with Riesz spaces valued measures and processes. We first study the lattice properties of processes of vector measures valued in an ordered vector space, but the main goal is to analyze those Banach lattice-valued processes of random variables, which include martingales, submartingales and supermartingales, that is an extension of the notion of asymptotic martingales to the infinite dimensional setting.
    [Show full text]
  • Semantics of Higher-Order Probabilistic Programs with Conditioning
    Semantics of higher-order probabilistic programs with conditioning Fredrik Dahlqvist Dexter Kozen University College London Cornell University [email protected] [email protected] Abstract—We present a denotational semantics for higher- exponentials. To accommodate conditioning and Bayesian order probabilistic programs in terms of linear operators between inference, we introduce ‘Bayesian types’, in which values are Banach spaces. Our semantics is rooted in the classical theory of decorated with a prior distribution. Based on this foundation, Banach spaces and their tensor products, but bears similarities with the well-known semantics of higher-order programs a` we give a type system and denotational semantics for an la Scott through the use ordered Banach spaces which allow idealized higher-order probabilistic language with sampling, definitions in terms of fixed points. Being based on a monoidal conditioning, and Bayesian inference. rather than cartesian closed structure, our semantics effectively We believe our approach should appeal to computer sci- treats randomness as a resource. entists, as it is true to traditional Scott-style denotational se- mantics (see IV-D). It should also appeal to mathematicians, I. INTRODUCTION statisticians and§ machine learning theorists, as it uses very Probabilistic programming has enjoyed a recent resurgence familiar mathematical objects from those fields. For example, of interest driven by new applications in machine learning a traditional perspective is that a Markov process is just a and statistical analysis of large datasets. The emergence of positive linear operator of norm 1 between certain Banach probabilistic programming languages such as Church and lattices [7, Ch. 19]. These are precisely the morphisms of our Anglican, which allow statisticians to construct and sample semantics.
    [Show full text]
  • MAXIMAL D-IDEALS in a RIESZ SPACE
    Can. J. Math., Vol. XXXV, No. 6, 1983, pp. 1010-1029 MAXIMAL d-IDEALS IN A RIESZ SPACE CHARLES B. HUIJSMANS AND BEN DE PAGTER 1. Introduction. We recall that the ideal / in an Archimedean Riesz space L is called a d-ideal whenever it follows from/ e / that {/} c /. Several authors (see [4], [5], [6], [12], [13], [15] and [18] ) have considered the class of all d-ideals in L, but the set^ of all maximal d-ideals in L has not been studied in detail in the literature. In [12] and [13] the present authors paid some attention to certain aspects of the theory of maximal d-ideals, however neglecting the fact that^, equipped with its hull-kernel topology, is a structure space of the underlying Riesz space L. The main purpose of the present paper is to investigate the topological properties of^ and to compare^ to other structure spaces of L, such as the space Jt of minimal prime ideals and the space Qe of all ^-maximal ideals in L (where e > 0 is a weak order unit). This study of^ makes it for instance possible to place some recent results of F. K. Dashiell, A. W. Hager and M. Henriksen [7], concerning the order completion of C(X) in the general setting of Riesz spaces. After establishing some preliminary results in Section 2, it is proved in Section 3 that for any uniformly complete Riesz space L with weak order unit the spaced is a compact quasi-F-space. This result makes it possible to use^ for the description of the order completion of L, which is done in Section 4.
    [Show full text]
  • The Grothendieck Property of Weak L Spaces and Marcinkiewicz Spaces
    TU DELFT MASTER THESIS APPLIED MATHEMATICS The Grothendieck property of Weak Lp spaces and Marcinkiewicz spaces Supervisor Author Prof. Dr. B. DE PAGTER P. VAN DEN BOSCH 28-1-2019 Introduction In this master thesis the proof of Lotz in [40] that Weak Lp spaces have the Grothendieck property is studied. The proof is slightly modified to be more explicit and easier to comprehend by introducing lemma’s to better separate different parts of the proof that more clearly reveal its structure. Fur- thermore, the more general Marcinkiewicz spaces are shown to sometimes have the Grothendieck property, using the sufficient conditions for a Banach lattice to have the Grothendieck property that Lotz derived in [40] to prove the Grothendieck property of Weak Lp spaces. For most of these conditions that together are sufficient, proving that Marcinkiewicz spaces satisfy them is done in a way very similar to the case of Weak Lp spaces. However, the proof of the (necessary) condition that the dual sometimes has order continuous norm does not allow for such a simple generalization and requires more work. Finally, by using some more recent results [19] about the existence of symmetric functionals in the dual, the conditions that are given for the Grothendieck property of Marcinkiewicz spaces are shown to be necessary, and we thereby obtain a characterization of the Marcinkiewicz spaces that have the Grothendieck property. Chapter 1 briefly summarizes well-known theory about Banach spaces required in the subsequent chapters: the weak topology, the Grothendieck property and characterizations thereof. In chapter 2 Banach lattices are introduced and Lotz’ sufficient conditions for the Grothendieck property of a Banach lattice are derived.
    [Show full text]
  • Real) Vector Spaces L and M,Andlett : L → M Be a Linear Operator
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 354, Number 5, Pages 2055{2077 S 0002-9947(01)02925-7 Article electronically published on December 27, 2001 THE SUPER ORDER DUAL OF AN ORDERED VECTOR SPACE AND THE RIESZ{KANTOROVICH FORMULA CHARALAMBOS D. ALIPRANTIS AND RABEE TOURKY Abstract. A classical theorem of F. Riesz and L. V. Kantorovich asserts that if L is a vector lattice and f and g are order bounded linear functionals on L, then their supremum (least upper bound) f _ g exists in L∼ and for each x 2 L it satisfies the so-called Riesz{Kantorovich formula: + f _ g (x)=sup f(y)+g(z): y; z 2 L+ and y + z = x : Related to the Riesz{Kantorovich formula is the following long-standing prob- lem: If the supremum of two order bounded linear functionals f and g on an ordered vector space exists, does it then satisfy the Riesz{Kantorovich for- mula? In this paper, we introduce an extension of the order dual of an ordered vector space and provide some answers to this long-standing problem. The ideas regarding the Riesz{Kantorovich formula owe their origins to the study of the fundamental theorems of welfare economics and the existence of com- petitive equilibrium. The techniques introduced here show that the existence of decentralizing prices for efficient allocations is closely related to the above- mentioned problem and to the properties of the Riesz{Kantorovich formula. 1. Introduction Consider two ordered (real) vector spaces L and M,andletT : L ! M be a linear operator.
    [Show full text]
  • Characterizations and Classifications of Some Classical Banach Spaces
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector ADVANCES IN MATHEMATICS 12, 367-401 (1974) Characterizations and Classifications of Some Classical Banach Spaces H. ELTON LACEY AND S. J. BERNAU* Department of Mathematics, University of Texas, ~tustin, Texas 78712 Communicated by E. Hewitt The classical Banach spaces L~(/~) and sublattices of C(T) are characterized in terms of their lattice structure. Isomorphic classifications of L~(/z) are also presented for the case of a finite measure. Also, some Banach space charac- terizations of L~(#) are developed. l. PRELIMINARIES The purpose of this paper is to present a survey of some structure theorems for certain classes of Banach spaces. These spaces are known as "classical Banach spaces." The spaces we are concerned with are of two distinct types, namely, Lp(/~) for 1 ~< p < oo and sublattices of C(T) for some compact Hausdorff space T. While most of the theorems in this paper have appeared in research papers, some of them are not widely known. Also, certain proofs have been heretofore inacessible and certain definitions not generally agreed upon (viz. the definition of an abstract L9 space). The discussion will of necessity include topics in the areas of measure theory, Boolean algebras, Banach spaces, Banach lattices, cardinal and ordinal numbers, and compact Hausdorff spaces. Such diversity makes it impossible for this paper to be completely self-contained. The reader is invited to check [11], [14], and [24] for any unexplained terminology or the necessary background theorems. We shall present a brief discussion of the parts of these areas which are used herein.
    [Show full text]
  • View Those Results As Characteriza- Tions of When the Identity Mapping Is Continuous with Respect to the Unbounded/Classical Convergences
    Unbounded convergences in vector lattices by Mitchell A. Taylor A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics Department of Mathematical and Statistical Sciences University of Alberta © Mitchell A. Taylor, 2018 Abstract Abstract. Suppose X is a vector lattice and there is a notion σ of convergence xα x in X. Then we can speak of an “un- −→ uσ bounded” version of this convergence by saying that xα x −−→ σ if xα x u 0 for every u X+. In the literature the | − | ∧ −→ ∈ unbounded versions of the norm, order and absolute weak con- vergence have been studied. Here we create a general theory of unbounded convergence, but with a focus on uo-convergence and those convergences deriving from locally solid topologies. We also give characterizations of minimal topologies in terms of unbounded topologies and uo-convergence. At the end we touch on the theory of bibases in Banach lattices. ii iii Preface The research in this thesis is an amalgamation of the papers Un- bounded topologies and uo-convergence in locally solid vector lattices, Metrizability of minimal and unbounded topologies, Completeness of unbounded convergences, and Extending topologies to the universal σ- completion of a vector lattice. The second paper was done in collabora- tion with Marko Kandi´c, and appeared in the Journal of Mathematical Analysis and Applications. The third paper is published in the Proceed- ings of the American Mathematical Society. There are also many new results in this thesis that do not appear in the aforementioned papers. The section on bibases in Banach lattices has not yet been submitted for publication, but there are plans to do so.
    [Show full text]
  • An Extension of the Concept of the Order Dual of a Riesz Space
    AN EXTENSION OF THE CONCEPT OF THE ORDER DUAL OF A RIESZ SPACE W. A. J. LUXEMBURG AND J. J. MASTERSON Introduction. Let L be a c-Dedekind complete Riesz space. In (8), H. Nakano uses an extension of the multiplication operator on a Riesz space into itself (analagous to the closed operator on a Hilbert space) to obtain a representation space for the Riesz space L. He calls such an operator a "dilata­ tor operator on L." More specifically, he shows that the set of all dilatator operators 35(£), when suitable operations are defined, is a Dedekind complete Riesz space which is isomorphic to the space of all functions defined and continuous on an open dense subset of some fixed totally disconnected Haus- dorfï space. The embedding of L in the function space is then obtained by showing that L is isomorphic to a Riesz subspace of £)(£). Moreover, when L is Dedekind complete, it is an ideal in 5)(L), and the topological space is extremally disconnected. Let L now be an arbitrary Riesz space. We shall call a linear functional <f> on L a normal integral whenever 0 < u7 \ 0 in L implies inf \<j>(u)\ = 0. (The terminology of Luxemburg and Zaanen in (4) will be adopted and knowledge of their series of notes on Banach Function Spaces, specifically (4, 5, 6, 7), will be assumed.) Ln~, the space of all such linear functionals, was introduced by H. Nakano (cf., (8)) and has been studied more recently by S. Kaplan (1, 2, 3) and, as mentioned above, by W.
    [Show full text]