Weak Topologies for the Closed Subsets of a Metrizable Space
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Weak Topologies
Weak topologies David Lecomte May 23, 2006 1 Preliminaries from general topology In this section, we are given a set X, a collection of topological spaces (Yi)i∈I and a collection of maps (fi)i∈I such that each fi maps X into Yi. We wish to define a topology on X that makes all the fi’s continuous. And we want to do this in the cheapest way, that is: there should be no more open sets in X than required for this purpose. −1 Obviously, all the fi (Oi), where Oi is an open set in Yi should be open in X. Then finite intersections of those should also be open. And then any union of finite intersections should be open. By this process, we have created as few open sets as required. Yet it is not clear that the collection obtained is closed under finite intersections. It actually is, as a consequence of the following lemma: Lemma 1 Let X be a set and let O ⊂ P(X) be a collection of subsets of X, such that • ∅ and X are in O; • O is closed under finite intersections. Then T = { O | O ⊂ O} is a topology on X. OS∈O Proof: By definition, T contains X and ∅ since those were already in O. Furthermore, T is closed under unions, again by definition. So all that’s left is to check that T is closed under finite intersections. Let A1 and A2 be two elements of T . Then there exist O1 and O2, subsets of O, such that A = O and A = O 1 [ 2 [ O∈O1 O∈O2 1 It is then easy to check by double inclusion that A ∩ A = O ∩ O 1 2 [ 1 2 O1∈O1 O2∈O2 Letting O denote the collection {O1 ∩ O2 | O1 ∈ O1 O2 ∈ O2}, which is a subset of O since the latter is closed under finite intersections, we get A ∩ A = O 1 2 [ O∈O This set belongs to T . -
Distinguished Property in Tensor Products and Weak* Dual Spaces
axioms Article Distinguished Property in Tensor Products and Weak* Dual Spaces Salvador López-Alfonso 1 , Manuel López-Pellicer 2,* and Santiago Moll-López 3 1 Department of Architectural Constructions, Universitat Politècnica de València, 46022 Valencia, Spain; [email protected] 2 Emeritus and IUMPA, Universitat Politècnica de València, 46022 Valencia, Spain 3 Department of Applied Mathematics, Universitat Politècnica de València, 46022 Valencia, Spain; [email protected] * Correspondence: [email protected] 0 Abstract: A local convex space E is said to be distinguished if its strong dual Eb has the topology 0 0 0 0 b(E , (Eb) ), i.e., if Eb is barrelled. The distinguished property of the local convex space Cp(X) of real- valued functions on a Tychonoff space X, equipped with the pointwise topology on X, has recently aroused great interest among analysts and Cp-theorists, obtaining very interesting properties and nice characterizations. For instance, it has recently been obtained that a space Cp(X) is distinguished if and only if any function f 2 RX belongs to the pointwise closure of a pointwise bounded set in C(X). The extensively studied distinguished properties in the injective tensor products Cp(X) ⊗# E and in Cp(X, E) contrasts with the few distinguished properties of injective tensor products related to the dual space Lp(X) of Cp(X) endowed with the weak* topology, as well as to the weak* dual of Cp(X, E). To partially fill this gap, some distinguished properties in the injective tensor product space Lp(X) ⊗# E are presented and a characterization of the distinguished property of the weak* dual of Cp(X, E) for wide classes of spaces X and E is provided. -
12 | Urysohn Metrization Theorem
12 | Urysohn Metrization Theorem In this chapter we return to the problem of determining which topological spaces are metrizable i.e. can be equipped with a metric which is compatible with their topology. We have seen already that any metrizable space must be normal, but that not every normal space is metrizable. We will show, however, that if a normal space space satisfies one extra condition then it is metrizable. Recall that a space X is second countable if it has a countable basis. We have: 12.1 Urysohn Metrization Theorem. Every second countable normal space is metrizable. The main idea of the proof is to show that any space as in the theorem can be identified with a subspace of some metric space. To make this more precise we need the following: 12.2 Definition. A continuous function i: X → Y is an embedding if its restriction i: X → i(X) is a homeomorphism (where i(X) has the topology of a subspace of Y ). 12.3 Example. The function i: (0; 1) → R given by i(x) = x is an embedding. The function j : (0; 1) → R given by j(x) = 2x is another embedding of the interval (0; 1) into R. 12.4 Note. 1) If j : X → Y is an embedding then j must be 1-1. 2) Not every continuous 1-1 function is an embedding. For example, take N = {0; 1; 2;::: } with the discrete topology, and let f : N → R be given ( n f n 0 if = 0 ( ) = 1 n if n > 0 75 12. -
General Topology
General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry). -
An Exploration of the Metrizability of Topological Spaces
AN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES DUSTIN HEDMARK Abstract. A study of the conditions under which a topological space is metrizable, concluding with a proof of the Nagata Smirnov Metrization The- orem Contents 1. Introduction 1 2. Product Topology 2 3. More Product Topology and R! 3 4. Separation Axioms 4 5. Urysohn's Lemma 5 6. Urysohn's Metrization Theorem 6 7. Local Finiteness and Gδ sets 8 8. Nagata-Smirnov Metrization Theorem 11 References 13 1. Introduction In this paper we will be exploring a basic topological notion known as metriz- ability, or whether or not a given topology can be understood through a distance function. We first give the reader some basic definitions. As an outline, we will be using these notions to first prove Urysohn's lemma, which we then use to prove Urysohn's metrization theorem, and we culminate by proving the Nagata Smirnov Metrization Theorem. Definition 1.1. Let X be a topological space. The collection of subsets B ⊂ X forms a basis for X if for any open U ⊂ X can be written as the union of elements of B Definition 1.2. Let X be a set. Let B ⊂ X be a collection of subsets of X. The topology generated by B is the intersection of all topologies on X containing B. Definition 1.3. Let X and Y be topological spaces. A map f : X ! Y . If f is a homeomorphism if it is a continuous bijection with a continuous inverse. If there is a homeomorphism from X to Y , we say that X and Y are homeo- morphic. -
DEFINITIONS and THEOREMS in GENERAL TOPOLOGY 1. Basic
DEFINITIONS AND THEOREMS IN GENERAL TOPOLOGY 1. Basic definitions. A topology on a set X is defined by a family O of subsets of X, the open sets of the topology, satisfying the axioms: (i) ; and X are in O; (ii) the intersection of finitely many sets in O is in O; (iii) arbitrary unions of sets in O are in O. Alternatively, a topology may be defined by the neighborhoods U(p) of an arbitrary point p 2 X, where p 2 U(p) and, in addition: (i) If U1;U2 are neighborhoods of p, there exists U3 neighborhood of p, such that U3 ⊂ U1 \ U2; (ii) If U is a neighborhood of p and q 2 U, there exists a neighborhood V of q so that V ⊂ U. A topology is Hausdorff if any distinct points p 6= q admit disjoint neigh- borhoods. This is almost always assumed. A set C ⊂ X is closed if its complement is open. The closure A¯ of a set A ⊂ X is the intersection of all closed sets containing X. A subset A ⊂ X is dense in X if A¯ = X. A point x 2 X is a cluster point of a subset A ⊂ X if any neighborhood of x contains a point of A distinct from x. If A0 denotes the set of cluster points, then A¯ = A [ A0: A map f : X ! Y of topological spaces is continuous at p 2 X if for any open neighborhood V ⊂ Y of f(p), there exists an open neighborhood U ⊂ X of p so that f(U) ⊂ V . -
Let H Be a Hilbert Space. on B(H), There Is a Whole Zoo of Topologies
Let H be a Hilbert space. On B(H), there is a whole zoo of topologies weaker than the norm topology – and all of them are considered when it comes to von Neumann algebras. It is, however, a good idea to concentrate on one of them right from the definition. My choice – and Murphy’s [Mur90, Chapter 4] – is the strong (or strong operator=STOP) topology: Definition. A von Neumann algebra is a ∗–subalgebra A ⊂ B(H) of operators acting nonde- generately(!) on a Hilbert space H that is strongly closed in B(H). (Every norm convergent sequence converges strongly, so A is a C∗–algebra.) This does not mean that one has not to know the other topologies; on the contrary, one has to know them very well, too. But it does mean that proof techniques are focused on the strong topology; if we use a different topology to prove something, then we do this only if there is a specific reason for doing so. One reason why it is not sufficient to worry only about the strong topology, is that the strong topology (unlike the norm topology of a C∗–algebra) is not determined by the algebraic structure alone: There are “good” algebraic isomorphisms between von Neumann algebras that do not respect their strong topologies. A striking feature of the strong topology on B(H) is that B(H) is order complete: Theorem (Vigier). If aλ λ2Λ is an increasing self-adjoint net in B(H) and bounded above (9c 2 B(H): aλ ≤ c8λ), then aλ converges strongly in B(H), obviously to its least upper bound in B(H). -
Chapter 14. Duality for Normed Linear Spaces
14.1. Linear Functionals, Bounded Linear Functionals, and Weak Topologies 1 Chapter 14. Duality for Normed Linear Spaces Note. In Section 8.1, we defined a linear functional on a normed linear space, a bounded linear functional, and the functional norm. In Proposition 8.1 (the proof is Exercise 8.2) it is shown that the collection of bounded linear functionals themselves form a normed linear space called the dual space of X, denoted X∗. In Chapters 14 and 15 we consider the mapping from X × X∗ → R defined by (x, ψ) 7→ ψ(x) to “uncover the analytic, geometric, and topological properties of Banach spaces.” The “departure point for this exploration” is the Hahn-Banach Theorem which is started and proved in Section 14.2 (Royden and Fitzpatrick, page 271). Section 14.1. Linear Functionals, Bounded Linear Functionals, and Weak Topologies Note. In this section we consider the linear space of all real valued linear function- als on linear space X (without requiring X to be named or the functionals to be bounded), denoted X]. We also consider a new topology on a normed linear space called the weak topology (the old topology which was induced by the norm we now may call the strong topology). For the deal X∗ of normed linear space X, the weak topology is called the weak-∗ topology. 14.1. Linear Functionals, Bounded Linear Functionals, and Weak Topologies 2 Note. Recall that if Y and Z are subspaces of a linear space then Y + Z is also a subspace of X (by Exercise 13.2) and that if Y ∩ Z = {0} then Y + Z is denoted T ⊕ Z and is called the direct sum of Y and Z. -
The Banach-Alaoglu Theorem for Topological Vector Spaces
The Banach-Alaoglu theorem for topological vector spaces Christiaan van den Brink a thesis submitted to the Department of Mathematics at Utrecht University in partial fulfillment of the requirements for the degree of Bachelor in Mathematics Supervisor: Fabian Ziltener date of submission 06-06-2019 Abstract In this thesis we generalize the Banach-Alaoglu theorem to topological vector spaces. the theorem then states that the polar, which lies in the dual space, of a neighbourhood around zero is weak* compact. We give motivation for the non-triviality of this theorem in this more general case. Later on, we show that the polar is sequentially compact if the space is separable. If our space is normed, then we show that the polar of the unit ball is the closed unit ball in the dual space. Finally, we introduce the notion of nets and we use these to prove the main theorem. i ii Acknowledgments A huge thanks goes out to my supervisor Fabian Ziltener for guiding me through the process of writing a bachelor thesis. I would also like to thank my girlfriend, family and my pet who have supported me all the way. iii iv Contents 1 Introduction 1 1.1 Motivation and main result . .1 1.2 Remarks and related works . .2 1.3 Organization of this thesis . .2 2 Introduction to Topological vector spaces 4 2.1 Topological vector spaces . .4 2.1.1 Definition of topological vector space . .4 2.1.2 The topology of a TVS . .6 2.2 Dual spaces . .9 2.2.1 Continuous functionals . -
Weak Topologies Weak-Type Topologies on Vector Spaces. Let X
Weak topologies Weak-type topologies on vector spaces. Let X be a vector space with the algebraic dual X]. Let Y ½ X] be a subspace. We want to de¯ne a topology σ on X in order to make continuous all elements of Y . Fix x0 2 X. If σ is such a topology, then the sets of the form x0 V";g := fx 2 X : jg(x) ¡ g(x0)j < "g = fx 2 X : jg(x ¡ x0)j < "g (" > 0; g 2 Y ) are open neighborhoods of x0. But this family is not a basis of σ-neighborhoods of x0, since the intersection of two of its members does not necessarily contain another member of the family. This is the reason why we instead consider the sets of the form x0 (1) V";g1;:::;gn := fx 2 X : jgi(x ¡ x0)j < "; i = 1; : : : ; ng (" > 0; n 2 N; gi 2 Y ) : x0 x0 It is easy to see that the intersection V \ V 0 of two of such sets ";g1;:::;gn " ;h1;:::;hm x0 00 0 contains V 00 where " = minf"; " g. " ;g1;:::;gn;h1;:::;hm Theorem 0.1. Let X be a vector space, and Y ½ X] a subspace which separates the points of X (that is, a so-called total subspace). 1. There exists a (unique) topology on X such that, for each x0 2 X, the sets (1) form a basis of neighborhoods of x0. This topology, denoted by σ(X; Y ), is called the weak topology determined by Y . 2. σ(X; Y ) is the weakest topology on X that makes continuous all elements of Y . -
Arxiv:Math/0405137V1 [Math.RT] 7 May 2004 Oino Disbecniuu Ersnain Fsc Group)
Draft: May 4, 2004 LOCALLY ANALYTIC VECTORS IN REPRESENTATIONS OF LOCALLY p-ADIC ANALYTIC GROUPS Matthew Emerton Northwestern University Contents 0. Terminology, notation and conventions 7 1. Non-archimedean functional analysis 10 2. Non-archimedean function theory 27 3. Continuous, analytic, and locally analytic vectors 43 4. Smooth, locally finite, and locally algebraic vectors 71 5. Rings of distributions 81 6. Admissible locally analytic representations 100 7. Representationsofcertainproductgroups 124 References 135 Recent years have seen the emergence of a new branch of representation theory: the theory of representations of locally p-adic analytic groups on locally convex p-adic topological vector spaces (or “locally analytic representation theory”, for short). Examples of such representations are provided by finite dimensional alge- braic representations of p-adic reductive groups, and also by smooth representations of such groups (on p-adic vector spaces). One might call these the “classical” ex- amples of such representations. One of the main interests of the theory (from the point of view of number theory) is that it provides a setting in which one can study p-adic completions of the classical representations [6], or construct “p-adic interpolations” of them (for example, by defining locally analytic analogues of the principal series, as in [20], or by constructing representations via the cohomology of arithmetic quotients of symmetric spaces, as in [9]). Locally analytic representation theory also plays an important role in the analysis of p-adic symmetric spaces; indeed, this analysis provided the original motivation for its development. The first “non-classical” examples in the theory were found by Morita, in his analysis of the p-adic upper half-plane (the p-adic symmetric arXiv:math/0405137v1 [math.RT] 7 May 2004 space attached to GL2(Qp)) [15], and further examples were found by Schneider and Teitelbaum in their analytic investigations of the p-adic symmetric spaces of GLn(Qp) (for arbitrary n) [19]. -
Orlicz - Pettis Theorems for Multiplier Convergent Operator Valued Series
Proyecciones Vol. 23, No 1, pp. 61-72, May 2004. Universidad Cat´olica del Norte Antofagasta - Chile ORLICZ - PETTIS THEOREMS FOR MULTIPLIER CONVERGENT OPERATOR VALUED SERIES CHARLES SWARTZ New Mexico State University, USA Received November 2003. Accepted March 2004. Abstract Let X, Y be locally convex spaces and L(X, Y ) the space of con- tinuous linear operators from X into Y . We consider 2 types of mul- tiplier convergent theorems for a series Tk in L(X, Y ).First,ifλ isascalarsequencespace,wesaythattheseries Tk is λ multiplier convergent for a locally convex topologyPτ on L(X, Y ) if the series P tkTk is τ convergent for every t = tk λ. We establish condi- tions on λ which guarantee that a λ multiplier{ } ∈ convergent series in theP weak or strong operator topology is λ multiplier convergent in the topology of uniform convergence on the bounded subsets of X.Sec- ond, we consider vector valued multipliers. If E is a sequence space of X valued sequences, the series Tk is E multiplier convergent in a locally convex topology η on Y if the series Tkxk is η convergent P for every x = xk E. We consider a gliding hump property on E { } ∈ P whichguaranteesthataseries Tk which is E multiplier convergent for the weak topology of Y is E multiplier convergent for the strong topology of Y . P 62 Charles Swartz 1. INTRODUCTION . The original Orlicz-Pettis Theorem which asserts that a series xk in anormedspace X which is subseries convergent in the weak topology of P X is actually subseries convergent in the norm topology of X can be inter- preted as a theorem about multiplier convergent series.