The Incompleteness of Weak Duals 1. Incompleteness of Weak Duals Of
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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. -
View of This, the Following Observation Should Be of Some Interest to Vector Space Pathologists
Can. J. Math., Vol. XXV, No. 3, 1973, pp. 511-524 INCLUSION THEOREMS FOR Z-SPACES G. BENNETT AND N. J. KALTON 1. Notation and preliminary ideas. A sequence space is a vector subspace of the space co of all real (or complex) sequences. A sequence space E with a locally convex topology r is called a K- space if the inclusion map E —* co is continuous, when co is endowed with the product topology (co = II^Li (R)*). A i£-space E with a Frechet (i.e., complete, metrizable and locally convex) topology is called an FK-space; if the topology is a Banach topology, then E is called a BK-space. The following familiar BK-spaces will be important in the sequel: ni, the space of all bounded sequences; c, the space of all convergent sequences; Co, the space of all null sequences; lp, 1 ^ p < °°, the space of all absolutely ^-summable sequences. We shall also consider the space <j> of all finite sequences and the linear span, Wo, of all sequences of zeros and ones; these provide examples of spaces having no FK-topology. A sequence space is solid (respectively monotone) provided that xy £ E whenever x 6 m (respectively m0) and y £ E. For x £ co, we denote by Pn(x) the sequence (xi, x2, . xn, 0, . .) ; if a i£-space (E, r) has the property that Pn(x) —» x in r for each x G E, then we say that (£, r) is an AK-space. We also write WE = \x Ç £: P„.(x) —>x weakly} and ^ = jx Ç E: Pn{x) —>x in r}. -
Bornologically Isomorphic Representations of Tensor Distributions
Bornologically isomorphic representations of distributions on manifolds E. Nigsch Thursday 15th November, 2018 Abstract Distributional tensor fields can be regarded as multilinear mappings with distributional values or as (classical) tensor fields with distribu- tional coefficients. We show that the corresponding isomorphisms hold also in the bornological setting. 1 Introduction ′ ′ ′r s ′ Let D (M) := Γc(M, Vol(M)) and Ds (M) := Γc(M, Tr(M) ⊗ Vol(M)) be the strong duals of the space of compactly supported sections of the volume s bundle Vol(M) and of its tensor product with the tensor bundle Tr(M) over a manifold; these are the spaces of scalar and tensor distributions on M as defined in [?, ?]. A property of the space of tensor distributions which is fundamental in distributional geometry is given by the C∞(M)-module isomorphisms ′r ∼ s ′ ∼ r ′ Ds (M) = LC∞(M)(Tr (M), D (M)) = Ts (M) ⊗C∞(M) D (M) (1) (cf. [?, Theorem 3.1.12 and Corollary 3.1.15]) where C∞(M) is the space of smooth functions on M. In[?] a space of Colombeau-type nonlinear generalized tensor fields was constructed. This involved handling smooth functions (in the sense of convenient calculus as developed in [?]) in par- arXiv:1105.1642v1 [math.FA] 9 May 2011 ∞ r ′ ticular on the C (M)-module tensor products Ts (M) ⊗C∞(M) D (M) and Γ(E) ⊗C∞(M) Γ(F ), where Γ(E) denotes the space of smooth sections of a vector bundle E over M. In[?], however, only minor attention was paid to questions of topology on these tensor products. -
7.3 Topological Vector Spaces, the Weak and Weak⇤ Topology on Banach Spaces
138 CHAPTER 7. ELEMENTS OF FUNCTIONAL ANALYSIS 7.3 Topological Vector spaces, the weak and weak⇤ topology on Banach spaces The following generalizes normed Vector space. Definition 7.3.1. Let X be a vector space over K, K = C, or K = R, and assume that is a topology on X. we say that X is a topological vector T space (with respect to ), if (X X and K X are endowed with the T ⇥ ⇥ respective product topology) +:X X X, (x, y) x + y is continuous ⇥ ! 7! : K X (λ, x) λ x is continuous. · ⇥ 7! · A topological vector space X is called locally convex,ifeveryx X has a 2 neighborhood basis consisting of convex sets, where a set A X is called ⇢ convex if for all x, y A, and 0 <t<1, it follows that tx +1 t)y A. 2 − 2 In order to define a topology on a vector space E which turns E into a topological vector space we (only) need to define an appropriate neighbor- hood basis of 0. Proposition 7.3.2. Assume that (E, ) is a topological vector space. And T let = U , 0 U . U0 { 2T 2 } Then a) For all x E, x + = x + U : U is a neighborhood basis of x, 2 U0 { 2U0} b) for all U there is a V so that V + V U, 2U0 2U0 ⇢ c) for all U and all R>0 there is a V ,sothat 2U0 2U0 λ K : λ <R V U, { 2 | | }· ⇢ d) for all U and x E there is an ">0,sothatλx U,forall 2U0 2 2 λ K with λ <", 2 | | e) if (E, ) is Hausdor↵, then for every x E, x =0, there is a U T 2 6 2U0 with x U, 62 f) if E is locally convex, then for all U there is a convex V , 2U0 2T with V U. -
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 . -
The Dual Topology for the Principal and Discrete Series on Semisimple Groupso
transactions of the american mathematical society Volume 152, December 1970 THE DUAL TOPOLOGY FOR THE PRINCIPAL AND DISCRETE SERIES ON SEMISIMPLE GROUPSO BY RONALD L. LIPSMAN Abstract. For a locally compact group G, the dual space G is the set of unitary equivalence classes of irreducible unitary representations equipped with the hull- kernel topology. We prove three results about G in the case that G is a semisimple Lie group: (1) the irreducible principal series forms a Hausdorff subspace of G; (2) the "discrete series" of square-integrable representations does in fact inherit the discrete topology from G; (3) the topology of the reduced dual Gr, that is the support of the Plancherel measure, is computed explicitly for split-rank 1 groups. 1. Introduction. It has been a decade since Fell introduced the hull-kernel topology on the dual space G of a locally compact group G. Although a great deal has been learned about the set G in many cases, little or no work has been done relating these discoveries to the topology. Recent work of Kazdan indicates that this is an unfortunate oversight. In this paper, we shall give some results for the case of a semisimple Lie group. Let G be a connected semisimple Lie group with finite center. Associated with a minimal parabolic subgroup of G there is a family of representations called the principal series. Our first result states roughly that if we strike out those members of the principal series that are not irreducible (a "thin" collection, actually), then what remains is a Hausdorff subspace of G. -
Appropriate Locally Convex Domains for Differential
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 86, Number 2, October 1982 APPROPRIATE LOCALLYCONVEX DOMAINS FOR DIFFERENTIALCALCULUS RICHARD A. GRAFF AND WOLFGANG M. RUESS Abstract. We make use of Grothendieck's notion of quasinormability to produce a comprehensive class of locally convex spaces within which differential calculus may be developed along the same lines as those employed within the class of Banach spaces and which include the previously known examples of such classes. In addition, we show that there exist Fréchet spaces which do not belong to any possible such class. 0. Introduction. In [2], the first named author introduced a theory of differential calculus in locally convex spaces. This theory differs from previous approaches to the subject in that the theory was an attempt to isolate a class of locally convex spaces to which the usual techniques of Banach space differential calculus could be extended, rather than an attempt to develop a theory of differential calculus for all locally convex spaces. Indeed, the original purpose of the theory was to study the maps which smooth nonlinear partial differential operators induce between Sobolev spaces by investigating the differentiability of these mappings with respect to a weaker (nonnormable) topology on the Sobolev spaces. The class of locally convex spaces thus isolated (the class of Z)-spaces, see Definition 1 below) was shown to include Banach spaces and several types of Schwartz spaces. A natural question to ask is whether there exists an easily-char- acterized class of D-spaces to which both of these classes belong. We answer this question in the affirmative in Theorem 1 below, the proof of which presents a much clearer picture of the nature of the key property of Z)-spaces than the corresponding result [2, Theorem 3.46]. -
Sufficient Generalities About Topological Vector Spaces
(November 28, 2016) Topological vector spaces Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ [This document is http://www.math.umn.edu/~garrett/m/fun/notes 2016-17/tvss.pdf] 1. Banach spaces Ck[a; b] 2. Non-Banach limit C1[a; b] of Banach spaces Ck[a; b] 3. Sufficient notion of topological vector space 4. Unique vectorspace topology on Cn 5. Non-Fr´echet colimit C1 of Cn, quasi-completeness 6. Seminorms and locally convex topologies 7. Quasi-completeness theorem 1. Banach spaces Ck[a; b] We give the vector space Ck[a; b] of k-times continuously differentiable functions on an interval [a; b] a metric which makes it complete. Mere pointwise limits of continuous functions easily fail to be continuous. First recall the standard [1.1] Claim: The set Co(K) of complex-valued continuous functions on a compact set K is complete with o the metric jf − gjCo , with the C -norm jfjCo = supx2K jf(x)j. Proof: This is a typical three-epsilon argument. To show that a Cauchy sequence ffig of continuous functions has a pointwise limit which is a continuous function, first argue that fi has a pointwise limit at every x 2 K. Given " > 0, choose N large enough such that jfi − fjj < " for all i; j ≥ N. Then jfi(x) − fj(x)j < " for any x in K. Thus, the sequence of values fi(x) is a Cauchy sequence of complex numbers, so has a limit 0 0 f(x). Further, given " > 0 choose j ≥ N sufficiently large such that jfj(x) − f(x)j < " .