DOCSLIB.ORG

Set-Valued Maps

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Set-Valued Maps Appendix A Set-Valued Maps Let X and Y be two nonempty sets. A set-valued map or multivalued map or point- to-set map or multifunction T W X ! 2Y from X to Y is a map that associates with any x 2 X a subset T.x/ of Y;thesetT.x/ is called the image of x under T.Theset Dom.T/ Dfx 2 X W T.x/ ¤;gis called the domain of T. Actually, a set-valued map T is characterized by its graph, the subset of X Y defined by Graph.T/ Df.x; y/ W y 2 T.x/g: Indeed, if A is a nonempty subset of the product space X Y, then the graph of a set-valued map T is defined by y 2 T.x/ if and only if .x; y/ 2 A: The domain of T is the projection of Graph.F/ on X.Theimage of T is a subset of Y defined by [ [ Im.T/ D T.x/ D T.x/: x2X x2Dom.T/ It is the projection of Graph.T/ on Y. A set-valued map T from X to Y is called strict if Dom.T/ D X, that is, if the image T.x/ is nonempty for all x 2 X.LetK be a nonempty subset of X and T be a strict set-valued map from K to Y.Itmaybe useful to extend it to the set-valued map TK from X to Y defined by T.x/; when x 2 K; T .x/ D K ;; when x … K; whose domain Dom.TK/ is K. © Springer International Publishing AG 2018 487 Q.H. Ansari et al., Vector Variational Inequalities and Vector Optimization, Vector Optimization, DOI 10.1007/978-3-319-63049-6 488 A Set-Valued Maps Let T be a set-valued map from X to Y and K  X, then we denote by TjK the restriction of T to K. Definition A.1 Let T W X ! 2Y be a set-valued map. For a nonempty subset A of X, we write [ T.A/ D T.x/: x2A If A D;, we write T.;/ D;.ThesetT.A/ is called the image of A under the set-valued map T. Y Theorem A.1 Let fA˛g˛2 be a family of nonempty subsets of X and T W X ! 2 be a set-valued map. (a) If A1  A2!then T.A1/  T.A2/; [ [ (b) T A˛ D T .A˛/; ˛2 ! ˛2 \ \ (c) T A˛ T .A˛/; ˛2 ˛2 (d) T .X n A1/ à T.X/ n T .A1/. Definition A.2 Let T1 and T2 be two set-valued maps from X to Y. •Theunion of T1 and T2 is a set-valued map .T1 [ T2/ from X to Y defined by .T1 [ T2/.x/ D T1.x/ [ T2.x/; for all x 2 X: •Theintersection of T1 and T2 is a set-valued map .T1 \ T2/ from X to Y defined by .T1 \ T2/.x/ D T1.x/ \ T2.x/; for all x 2 X: •TheCartesian product of T1 and T2 is a set-valued map .T1 T2/ from X to Y Y defined by .T1 T2/.x/ D T1.x/ T2.x/; for all x 2 X: •IfT1 is a set-valued map from X to Y and T2 is another set-valued map from Y to Z, then the composition product of T2 by T1 is a set-valued map .T2 ı T1/ from X to Z defined by .T2 ı T1/.x/ D T2.T1.x//; for all x 2 X: A Set-Valued Maps 489 Theorem A.2 Let T1 and T2 be set-valued maps from X to Y and A be a nonempty subset of X. Then (a) .T1 [ T2/.A/ D T1.A/ [ T2.A/; (b) .T1 \ T2/.A/  T1.A/ \ T2.A/; (c) .T1 T2/.A/  T1.A/ T2.A/; (d) .T2 ı T1/.A/ D T2.T1.A//: Definition A.3 If T is a set-valued map from X to Y, then the inverse T1 of T is defined by T1. y/ Dfx 2 X W y 2 T.x/g; for all y 2 Y Further, let B be a subset of Y.Theupper inverse image T1.B/ and lower inverse 1 image TC .B/ of B under F are defined by T1.B/ Dfx 2 X W T.x/ \ B ¤;g and 1 TC .B/ Dfx 2 X W T.x/  Bg: 1 1 We also write T .;/ D;and TC .;/ D;. It is clear from the definition of inverse of T that .T1/1 D T and y 2 T.x/ if and only if x 2 T1. y/. We have the following relations between domain, graph and image of T and T1. Dom.T1/ D Im.T/; Im.T1/ D Dom.T/ and Graph.T1/ Df. y; x/ 2 Y X W .x; y/ 2 Graph.T/g: Theorem A.3 Let fB˛g˛2 be a family of nonempty subsets of Y, A  X and B  Y. Let T W X ! 2Y be a set-valued map. 1 1 (a) If B1  B2,thenT .B1/  T .B2/; 1 (b) A TC .T.A//; . 1. // (c) B T TC B! ; [ [ 1 1 (d) TC B˛ TC .B˛/; ˛2 ! ˛2 \ \ 1 1 (e) TC B˛ D TC .B˛/; ˛2 ˛2 (f) T1.T.A// A; 490 A Set-Valued Maps ! \ \ 1 1 (g) T B˛ T .B˛/; ˛2 ! ˛2 [ [ 1 1 (h) T B˛ D T .B˛/. ˛2 ˛2 Y Theorem A.4 Let T1; T2 W X ! 2 be set-valued maps such that .T1 \ T2/.x/ ¤; for all x 2 X and B  Y. Then 1 1 1 (a) .T1 [ T2/ .B/ D T1 .B/ [ T2 .B/; 1 1 1 (b) .T1 \ T2/ .B/ T1 .B/ \ T2 .B/; 1 1 1 (c) .T1 [ T2/C .B/ D T1C.B/ \ T2C.B/; 1 1 1 (d) .T1 \ T2/C .B/ T1C.B/ [ T2C.B/; Y Z Theorem A.5 Let T1 W X ! 2 and T2 W Y ! 2 be set-valued maps. Then for any set B  Z, we have . /1 . / 1 1. / (a) T2 ı T1 C B D T1C T2C B ; 1 1 1 (b) .T2 ı T1/ .B/ D T1 T2 .B/ . Y Z Theorem A.6 Let T1 W X ! 2 and T2 W X ! 2 be set-valued maps. Then for any sets B  Y and D  Z, we have 1 1 1 (a) .T1 T2/C .B D/ D T1C.B/ \ T2C.D/; 1 1 1 (b) .T1 T2/ .B D/ D T1 .B/ \ T2 .D/. For further details and applications of set-valued maps, we refer to [1–8]andthe references therein. Appendix B Some Algebraic Concepts Let K and D be nonempty subsets of a vector space X.Thealgebraic sum and algebraic difference of K and D are, respectively, defined as K C D Dfx C y W x 2 K and y 2 Dg and K D Dfx y W x 2 K and y 2 Dg: Let be any real number, then K is defined as K Dfx W x 2 Kg: We pointed out that K C K D 2K is not true in general for any nonempty subset K of a vector space X. Definition B.1 Let X be a vector space. The space of all linear mappings from X to R is called algebraic dual of X and it is denoted by X0. Definition B.2 A subset K of a vector space X is called • balanced if it is nonempty and ˛K K for all ˛ 2 Œ1; 1; • absolutely convex if it is convex and balanced; • absorbing or absorbent if for each x 2 X, there exists >0such that x 2 K for all jjÄ. Note that an absorbing set contains the zero of X. Theorem B.1 Let X and Y be vector spaces and f W X ! Y be a linear map. (a) If K is a balanced subset of X, then f .K/ is balanced. (b) If K is absorbing and f is onto, then f .K/ is absorbing. (c) Inverse image under f of absorbing or balanced subsets of Y are absorbing or balanced, respectively, subset of X. © Springer International Publishing AG 2018 491 Q.H. Ansari et al., Vector Variational Inequalities and Vector Optimization, Vector Optimization, DOI 10.1007/978-3-319-63049-6 492 B Some Algebraic Concepts Definition B.3 Let K be a nonempty subset of a vector space X. • The set cor.K/ Dfy 2 K W for every x 2 X there is a >0N with y C x 2 C for all 2 Œ0; N g is called the algebraic interior of K (or the core of K). •ThesetK is called algebraic open if K D cor.K/. • The set of all elements of X which neither belong to cor.K/ nor to cor.X n K/ is called the algebraic boundary of K. •Anelementx 2 X is called linearly accessible from K if there exists y 2 K, y ¤ x, such that y C .1 /x 2 K for all 2 .0; 1. The union of K and the set of all linearly accessible elements from K is called the algebraic closure of K and it is denoted by lin.K/ D K [fx 2 X W x is linearly accessible from Kg: The set K is called algebraic closed if K D lin.K/.
Load more

Recommended publications
  • Topological Vector Spaces and Algebras
    Joseph Muscat 2015 1 Topological Vector Spaces and Algebras [email protected] 1 June 2016 1 Topological Vector Spaces over R or C Recall that a topological vector space is a vector space with a T0 topology such that addition and the field action are continuous. When the field is F := R or C, the field action is called scalar multiplication. Examples: A N • R , such as sequences R , with pointwise convergence. p • Sequence spaces ℓ (real or complex) with topology generated by Br = (a ): p a p < r , where p> 0. { n n | n| } p p p p • LebesgueP spaces L (A) with Br = f : A F, measurable, f < r (p> 0). { → | | } R p • Products and quotients by closed subspaces are again topological vector spaces. If π : Y X are linear maps, then the vector space Y with the ini- i → i tial topology is a topological vector space, which is T0 when the πi are collectively 1-1. The set of (continuous linear) morphisms is denoted by B(X, Y ). The mor- phisms B(X, F) are called ‘functionals’. +, , Finitely- Locally Bounded First ∗ → Generated Separable countable Top. Vec. Spaces ///// Lp 0 <p< 1 ℓp[0, 1] (ℓp)N (ℓp)R p ∞ N n R 2 Locally Convex ///// L p > 1 L R , C(R ) R pointwise, ℓweak Inner Product ///// L2 ℓ2[0, 1] ///// ///// Locally Compact Rn ///// ///// ///// ///// 1. A set is balanced when λ 6 1 λA A. | | ⇒ ⊆ (a) The image and pre-image of balanced sets are balanced. ◦ (b) The closure and interior are again balanced (if A 0; since λA = (λA)◦ A◦); as are the union, intersection, sum,∈ scaling, T and prod- uct A ⊆B of balanced sets.
    [Show full text]
  • Functional Analysis 1 Winter Semester 2013-14
    Functional analysis 1 Winter semester 2013-14 1. Topological vector spaces Basic notions. Notation. (a) The symbol F stands for the set of all reals or for the set of all complex numbers. (b) Let (X; τ) be a topological space and x 2 X. An open set G containing x is called neigh- borhood of x. We denote τ(x) = fG 2 τ; x 2 Gg. Definition. Suppose that τ is a topology on a vector space X over F such that • (X; τ) is T1, i.e., fxg is a closed set for every x 2 X, and • the vector space operations are continuous with respect to τ, i.e., +: X × X ! X and ·: F × X ! X are continuous. Under these conditions, τ is said to be a vector topology on X and (X; +; ·; τ) is a topological vector space (TVS). Remark. Let X be a TVS. (a) For every a 2 X the mapping x 7! x + a is a homeomorphism of X onto X. (b) For every λ 2 F n f0g the mapping x 7! λx is a homeomorphism of X onto X. Definition. Let X be a vector space over F. We say that A ⊂ X is • balanced if for every α 2 F, jαj ≤ 1, we have αA ⊂ A, • absorbing if for every x 2 X there exists t 2 R; t > 0; such that x 2 tA, • symmetric if A = −A. Definition. Let X be a TVS and A ⊂ X. We say that A is bounded if for every V 2 τ(0) there exists s > 0 such that for every t > s we have A ⊂ tV .
    [Show full text]
  • General Reflexivity for Absolutely Convex Sets
    University of New Hampshire University of New Hampshire Scholars' Repository Doctoral Dissertations Student Scholarship Spring 2019 General Reflexivity orF Absolutely Convex Sets Mahtab Lak University of New Hampshire, Durham Follow this and additional works at: https://scholars.unh.edu/dissertation Recommended Citation Lak, Mahtab, "General Reflexivity For Absolutely Convex Sets" (2019). Doctoral Dissertations. 2454. https://scholars.unh.edu/dissertation/2454 This Dissertation is brought to you for free and open access by the Student Scholarship at University of New Hampshire Scholars' Repository. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of University of New Hampshire Scholars' Repository. For more information, please contact [email protected]. GENERAL REFLEXIVITY FOR ABSOLUTELY CONVEX SETS BY MAHTAB LAK BS, Applied Mathematics, Sheikh-Bahaee University of Isfahan, Iran, September 2008 MS, Mathematics, Isfahan University of Technology, Iran , 2012 DISSERTATION Submitted to the University of New Hampshire in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics May 2019 ALL RIGHTS RESERVED ©2019 Mahtab Lak This dissertation has been examined and approved in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics by: Dissertation Director, Don Hadwin, Professor of Mathematics and Statistics Eric Nordgren, Professor of Mathematics and Statistics Junhoa Shen, Professor of Mathematics and Statistics John Gibson, Associate Professor of Mathematics and Statistics Mehment Orhan, Senior Lecture of Mathematics and Statistics on 4/15/2019. Original approval signatures are on file with the University of New Hampshire Graduate School. iii ACKNOWLEDGMENTS Firstly, I would like to express my sincere gratitude to my advisor Prof.
    [Show full text]
  • Nonlinear Monotone Operators with Values in 9(X, Y)
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 140, 83-94 (1989) Nonlinear Monotone Operators with Values in 9(X, Y) N. HADJISAVVAS, D. KRAVVARITIS, G. PANTELIDIS, AND I. POLYRAKIS Department of Mathematics, National Technical University of Athens, Zografou Campus, 15773 Athens, Greece Submitted by C. Foias Received April 2, 1987 1. INTRODUCTION Let X be a topological vector space, Y an ordered topological vector space, and 9(X, Y) the space of all linear and continuous mappings from X into Y. If T is an operator from X into 9(X, Y) (generally multivalued) with domain D(T), T is said to be monotone if for all xieD(T) and Aie T(x,), i= 1, 2. In the scalar case Y = R, this definition coincides with the well known definition of monotone operator (cf. [ 12,4]). An important subclass of monotone operators T: X+ 9(X, Y) consists of the subdifferentials of convex operators from X into Y, which have been studied by Valadier [ 171, Kusraev and Kutateladze [ 111, Papageorgiou [ 131, and others. Some properties of monotone operators have been investigated by Kirov [7, 81, in connection with the study of the differentiability of convex operators. In this paper we begin our investigation of the properties of monotone operators from X into 3(X, Y) by introducing, in Section 3, the notion of hereditary order convexity of subsets of 9(X, Y). This notion plays a key role in our discussion and expresses a kind of separability between sets and points, Its interest relies on the fact that the images of maximal monotone operators are hereditarily order-convex.
    [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]
  • On Barycenters of Probability Measures
    On barycenters of probability measures Sergey Berezin Azat Miftakhov Abstract A characterization is presented of barycenters of the Radon probability measures supported on a closed convex subset of a given space. A case of particular interest is studied, where the underlying space is itself the space of finite signed Radon measures on a metric compact and where the corresponding support is the convex set of probability measures. For locally compact spaces, a simple characterization is obtained in terms of the relative interior. 1. The main goal of the present note is to characterize barycenters of the Radon probability measures supported on a closed convex set. Let X be a Fr´echet space. Without loss of generality, the topology on X is generated by the translation-invariant metric ρ on X (for details see [2]). We denote the set of Radon probability measures on X by P(X). By definition, the barycen- ter a ∈ X of a measure µ ∈ P(X) is a = xµ(dx) (1) XZ if the latter integral exists in the weak sense, that is, if the following holds Λa = Λxµ(dx) (2) XZ for all Λ ∈ X∗, where X∗ is the topological dual of X. More details on such integrals can be found in [2, Chapter 3]. Note that if (1) exists, then a = xµ(dx), (3) suppZ µ arXiv:1911.07680v2 [math.PR] 20 Apr 2020 and, by the Hahn–Banach separation theorem, one has a ∈ co(supp µ), where co(·) stands for the convex hull. From now on, we will use the bar over a set in order to denote its topological closure.
    [Show full text]
  • Some Special Sets in an Exponential Vector Space
    Some Special Sets in an Exponential Vector Space Priti Sharma∗and Sandip Jana† Abstract In this paper, we have studied ‘absorbing’ and ‘balanced’ sets in an Exponential Vector Space (evs in short) over the field K of real or complex. These sets play pivotal role to describe several aspects of a topological evs. We have characterised a local base at the additive identity in terms of balanced and absorbing sets in a topological evs over the field K. Also, we have found a sufficient condition under which an evs can be topologised to form a topological evs. Next, we have introduced the concept of ‘bounded sets’ in a topological evs over the field K and characterised them with the help of balanced sets. Also we have shown that compactness implies boundedness of a set in a topological evs. In the last section we have introduced the concept of ‘radial’ evs which characterises an evs over the field K up to order-isomorphism. Also, we have shown that every topological evs is radial. Further, it has been shown that “the usual subspace topology is the finest topology with respect to which [0, ∞) forms a topological evs over the field K”. AMS Subject Classification : 46A99, 06F99. Key words : topological exponential vector space, order-isomorphism, absorbing set, balanced set, bounded set, radial evs. 1 Introduction Exponential vector space is a new algebraic structure consisting of a semigroup, a scalar arXiv:2006.03544v1 [math.FA] 5 Jun 2020 multiplication and a compatible partial order which can be thought of as an algebraic axiomatisation of hyperspace in topology; in fact, the hyperspace C(X ) consisting of all non-empty compact subsets of a Hausdörff topological vector space X was the motivating example for the introduction of this new structure.
    [Show full text]
  • Spectral Radii of Bounded Operators on Topological Vector Spaces
    SPECTRAL RADII OF BOUNDED OPERATORS ON TOPOLOGICAL VECTOR SPACES VLADIMIR G. TROITSKY Abstract. In this paper we develop a version of spectral theory for bounded linear operators on topological vector spaces. We show that the Gelfand formula for spectral radius and Neumann series can still be naturally interpreted for operators on topological vector spaces. Of course, the resulting theory has many similarities to the conventional spectral theory of bounded operators on Banach spaces, though there are several im- portant differences. The main difference is that an operator on a topological vector space has several spectra and several spectral radii, which fit a well-organized pattern. 0. Introduction The spectral radius of a bounded linear operator T on a Banach space is defined by the Gelfand formula r(T ) = lim n T n . It is well known that r(T ) equals the n k k actual radius of the spectrum σ(T ) p= sup λ : λ σ(T ) . Further, it is known −1 {| | ∈ } ∞ T i that the resolvent R =(λI T ) is given by the Neumann series +1 whenever λ − i=0 λi λ > r(T ). It is natural to ask if similar results are valid in a moreP general setting, | | e.g., for a bounded linear operator on an arbitrary topological vector space. The author arrived to these questions when generalizing some results on Invariant Subspace Problem from Banach lattices to ordered topological vector spaces. One major difficulty is that it is not clear which class of operators should be considered, because there are several non-equivalent ways of defining bounded operators on topological vector spaces.
    [Show full text]
  • Locally Convex Spaces Manv 250067-1, 5 Ects, Summer Term 2017 Sr 10, Fr
    LOCALLY CONVEX SPACES MANV 250067-1, 5 ECTS, SUMMER TERM 2017 SR 10, FR. 13:15{15:30 EDUARD A. NIGSCH These lecture notes were developed for the topics course locally convex spaces held at the University of Vienna in summer term 2017. Prerequisites consist of general topology and linear algebra. Some background in functional analysis will be helpful but not strictly mandatory. This course aims at an early and thorough development of duality theory for locally convex spaces, which allows for the systematic treatment of the most important classes of locally convex spaces. Further topics will be treated according to available time as well as the interests of the students. Thanks for corrections of some typos go out to Benedict Schinnerl. 1 [git] • 14c91a2 (2017-10-30) LOCALLY CONVEX SPACES 2 Contents 1. Introduction3 2. Topological vector spaces4 3. Locally convex spaces7 4. Completeness 11 5. Bounded sets, normability, metrizability 16 6. Products, subspaces, direct sums and quotients 18 7. Projective and inductive limits 24 8. Finite-dimensional and locally compact TVS 28 9. The theorem of Hahn-Banach 29 10. Dual Pairings 34 11. Polarity 36 12. S-topologies 38 13. The Mackey Topology 41 14. Barrelled spaces 45 15. Bornological Spaces 47 16. Reflexivity 48 17. Montel spaces 50 18. The transpose of a linear map 52 19. Topological tensor products 53 References 66 [git] • 14c91a2 (2017-10-30) LOCALLY CONVEX SPACES 3 1. Introduction These lecture notes are roughly based on the following texts that contain the standard material on locally convex spaces as well as more advanced topics.
    [Show full text]
  • A NOTE on GENERALIZED INTERIORS of SETS Let X Be A
    DIDACTICA MATHEMATICA, Vol. 31(2013), No 1, pp. 37{42 A NOTE ON GENERALIZED INTERIORS OF SETS Anca Grad Abstract. This article familiarizes the reader with generalized interiors of sets, as they are extremely useful when formulating optimality conditions for problems defined on topological vector spaces. Two original results connecting the quasi- relative interior and the quasi interior are included. MSC 2000. 49K99, 90C46 Key words. convex sets, algebraic interior, topological interior, quasi interior, quasi-relative interior 1. INTRODUCTION Let X be a topological vector space and let X∗ be the topological dual space of X. Given a linear continuous functional x∗ 2 X∗ and a point x 2 X, we denote by hx∗; xi the value of x∗ at x. The normal cone associated with a set M ⊆ X is defined by fx∗ 2 X∗ : hx∗; y − xi ≤ 0 for all y 2 Mg if x 2 M N (x) := M ; otherwise. Given a nonempty cone C ⊆ X, its dual cone is the set C+ := fx∗ 2 X∗ : hx∗; xi ≥ 0 for all x 2 Cg: We also use the following subset of C+: (1) C+0 := fx∗ 2 X∗ : hx∗; xi > 0 for all x 2 Cnf0gg: 2. GENERALIZED INTERIORS OF SETS. DEFINITIONS AND BASIC PROPERTIES Interior and generalized interior notions of sets are highly important when tackling optimization problems, as they are often used in formulating regular- ity conditions. Let X be a nontrivial vector space, and let M ⊆ X be a set. The algebraic interior of M is defined by x 2 X : for each y 2 X there exists δ > 0 such that core M := : x + λy 2 M for all λ 2 [0; δ] The intrinsic core of M is defined by x 2 X : for each y 2 aff(M − M) there exists δ > 0 icr M := such that x + λy 2 M for all λ 2 [0; δ] 38 Anca Grad 2 (see Holmes G.
    [Show full text]
  • Arxiv:2105.06358V1 [Math.FA] 13 May 2021 Xml,I 2 H.3,P 31.I 7,Qudfie H Ocp Fa of Concept the [7]) Defined in Qiu Complete [7], Quasi-Fast for in As See (Denoted 1371]
    INDUCTIVE LIMITS OF QUASI LOCALLY BAIRE SPACES THOMAS E. GILSDORF Department of Mathematics Central Michigan University Mt. Pleasant, MI 48859 USA [email protected] May 14, 2021 Abstract. Quasi-locally complete locally convex spaces are general- ized to quasi-locally Baire locally convex spaces. It is shown that an inductive limit of strictly webbed spaces is regular if it is quasi-locally Baire. This extends Qiu’s theorem on regularity. Additionally, if each step is strictly webbed and quasi- locally Baire, then the inductive limit is quasi-locally Baire if it is regular. Distinguishing examples are pro- vided. 2020 Mathematics Subject Classification: Primary 46A13; Sec- ondary 46A30, 46A03. Keywords: Quasi locally complete, quasi-locally Baire, inductive limit. arXiv:2105.06358v1 [math.FA] 13 May 2021 1. Introduction and notation. Inductive limits of locally convex spaces have been studied in detail over many years. Such study includes properties that would imply reg- ularity, that is, when every bounded subset in the in the inductive limit is contained in and bounded in one of the steps. An excellent introduc- tion to the theory of locally convex inductive limits, including regularity properties, can be found in [1]. Nevertheless, determining whether or not an inductive limit is regular remains important, as one can see for example, in [2, Thm. 34, p. 1371]. In [7], Qiu defined the concept of a quasi-locally complete space (denoted as quasi-fast complete in [7]), in 1 2 THOMASE.GILSDORF which each bounded set is contained in abounded set that is a Banach disk in a coarser locally convex topology, and proves that if an induc- tive limit of strictly webbed spaces is quasi-locally complete, then it is regular.
    [Show full text]
  • Arxiv:2107.04662V1 [Math.GN]
    ON LINEAR CONTINUOUS OPERATORS BETWEEN DISTINGUISHED SPACES Cp(X) JERZY KA¸KOL AND ARKADY LEIDERMAN Abstract. As proved in [16], for a Tychonoff space X, a locally convex space Cp(X) is distinguished if and only if X is a ∆-space. If there exists a linear con- tinuous surjective mapping T : Cp(X) → Cp(Y ) and Cp(X) is distinguished, then Cp(Y ) also is distinguished [17]. Firstly, in this paper we explore the following question: Under which con- ditions the operator T : Cp(X) → Cp(Y ) above is open? Secondly, we devote a special attention to concrete distinguished spaces Cp([1, α]), where α is a countable ordinal number. A complete characterization of all Y which admit a linear continuous surjective mapping T : Cp([1, α]) → Cp(Y ) is given. We also observe that for every countable ordinal α all closed linear subspaces of Cp([1, α]) are distinguished, thereby answering an open question posed in [17]. Using some properties of ∆-spaces we prove that a linear continuous sur- jection T : Cp(X) → Ck(X)w, where Ck(X)w denotes the Banach space C(X) endowed with its weak topology, does not exist for every infinite metrizable compact C-space X (in particular, for every infinite compact X ⊂ Rn). 1. Introduction ′ A locally convex space (lcs) E is called distinguished if its strong dual Eβ = (E′, β(E′, E)) is a barrelled space. A. Grothendieck [11] proved that a metrizable ′ lcs is distinguished if and only if Eβ is bornological. Also, if all bounded subsets of ′ the strong dual Eβ of a metrizable lcs are metrizable, then E is distinguished [11].
    [Show full text]