DOCSLIB.ORG

Noncomplete Mackey Topologies on Banach Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Noncomplete Mackey Topologies on Banach Spaces Bull. Aust. Math. Soc. 81 (2010), 409–413 doi:10.1017/S0004972709001154 NONCOMPLETE MACKEY TOPOLOGIES ON BANACH SPACES JOSÉ BONET ˛ and BERNARDO CASCALES (Received 28 March 2009) Abstract Answering a question of W. Arendt and M. Kunze in the negative, we construct a Banach space X and a norm closed weak* dense subspace Y of the dual X 0 of X such that X, endowed with the Mackey topology µ.X; Y / of the dual pair hX; Y i, is not complete. 2000 Mathematics subject classification: primary 46B10; secondary 46B50, 46A03. Keywords and phrases: Banach spaces, Mackey topologies, norming subspaces, Krein–Smulyan theorem. The following problem appeared in a natural way in connection with the study of Pettis integrability with respect to norming subspaces developed by Kunze in his PhD thesis T5U. This question was put to the authors by Kunze himself and his thesis advisor W. Arendt. PROBLEM. Suppose that .X; k·k/ is a Banach space and Y is a subspace of its topological dual X 0 which is norm closed and weak* dense. Is there a complete topology of the dual pair hX; Y i in X? We use the locally convex space (lcs) notation as in T4, 6, 7U. In particular, σ .X; Y / and µ.X; Y / denote the weak topology and the Mackey topology in X associated with the dual pair hX; Y i. For a Banach space X with topological dual X 0, the weak* topology is σ .X 0; X/. By the Bourbaki–Robertson lemma T4, Section 18.4.4U there is a complete topology in X of the dual pair hX; Y i if and only if the space .X; µ.X; Y // is complete. Therefore, the original question and the following are equivalent. PROBLEM A. Let .X; k·k/ be a Banach space. Is .X; µ.X; Y // complete for every norm closed weak* dense subspace Y of the dual space X 0? Let .X; k·k/ be a normed space. A subspace Y of X 0 is said to be norming if 0 0 the function p of X given by p.x/ D supfjx .x/j V x 2 Y \ BX0 g is a norm equivalent The research of Bonet was partially supported by FEDER and MEC Project MTM2007-62643 and by GV Prometeo/2008/101. The research of Cascales was supported by FEDER and MEC Project MTM2008- 05396 and by Fundacion´ Seneca´ de la CARM, project 08848/PI/08. c 2010 Australian Mathematical Publishing Association Inc. 0004-9727/2010 $16.00 409 Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.76, on 02 Oct 2021 at 11:54:59, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972709001154 410 J. Bonet and B. Cascales [2] to k·k. We notice that Problem A is not affected by changing the given norm of X to any equivalent one. Thus, to study Problem A for some norming subspace Y ⊂ X 0 we can and will assume that Y is indeed 1-norming, that is, kxk D supfjx0.x/j V x0 2 Y \ BX0 g. Let us observe that under the conditions of Problem A, if .X; µ.X; Y // is quasi-complete (in particular, complete), then the Krein–Smulyan theorem (see T4, Section 24.5(4)U) implies that for every σ .X; Y /-compact subset H of X, the σ .X; Y /- σ .X;Y / closed absolutely convex hull M VD aco H of H is also σ .X; Y /-compact. There are several papers dealing with the validity of the Krein–Smulyan theorem for topologies weaker than the weak topology; see, for instance, T1, 2U where it is proved that for every Banach space X not containing `1.T0; 1U/ and for every 1-norming subspace Y ⊂ X 0, if H is a norm bounded σ .X; Y /-compact subset of σ .X;Y / X then aco H is σ .X; Y /-compact. It was proved in T3U that the hypothesis `1.T0; 1U/ 6⊂ X is also necessary for the latter. The following useful observation will be used several times later. PROPOSITION 1. Let .X; k·k/ be a Banach space and let Y be a 1-norming subspace of X 0. If .X; µ.X; Y // is quasi-complete, then every σ .X; Y /-compact subset of X is norm bounded. PROOF. Let H ⊂ X be σ .X; Y /-compact. As already noted, the Krein–Smulyan theorem T4, Section 24.5(4)U implies that the σ .X; Y /-closed absolutely convex hull σ .X;Y / M VD aco H is σ .X; Y /-compact. Therefore, M is an absolutely convex, bounded and complete subset of the lcs .X; σ .X; Y //. Now we can apply T4, Section 20.11(2)U to obtain that M is a Banach disc, that is, X VD S nM is a M n2N Banach space with the norm kxkM VD inffλ ≥ 0 V x 2 λMg; x 2 X M : Since M is bounded in .X; σ .X; Y //, the inclusion J V X M ! .X; σ .X; Y // is continuous, therefore J V X M ! .X; k·k/ has a closed graph, hence it is continuous by the closed graph theorem. In particular, the image of the closed unit ball M of X M is bounded in .X; k·k/, and the proof is complete. 2 The following example is an immediate consequence of the foregoing. EXAMPLE 2. Let X D C.T0; 1U/ be endowed with its supremum norm and take 0 Y VD spanfδx V x 2 T0; 1Ug ⊂ X : Then .X; µ.X; Y // is not quasi-complete. PROOF. Notice that σ .X; Y / coincides with the topology τp of pointwise convergence on C.T0; 1U/. Since there are sequences τp-convergent to zero which are not norm bounded, .X; µ.X; Y // cannot be quasi-complete by Proposition 1. 2 Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.76, on 02 Oct 2021 at 11:54:59, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972709001154 [3] Noncomplete Mackey topologies on Banach spaces 411 The subspace Y of X 0 in Example 2 is weak* dense in X 0 but not norm closed. Another example of the same nature is the following: take X D c0, Y D ', the space of 0 sequences with finitely many nonzero coordinates, which is norm dense in X D `1. In this case µ.X; Y / D σ .X; Y /, since every absolutely convex σ .Y; X/-compact subset of Y is finite-dimensional by the Baire category theorem. In this case .X; σ .X; Y // is not even sequentially complete. The following example, taken from T3, Lemma 11U, provides the negative solution to Problem A. 1 EXAMPLE 3. Take X D .` .T0; 1U/; k·k1/ and consider the space Y D C.T0; 1U/ of continuous functions on T0; 1U as a norming subspace of the dual X 0 D `1.T0; 1U/. Then .X; µ.X; Y // is not quasi-complete. 1 PROOF. Let H VD fex V x 2 T0; 1Ug be the canonical basis of ` .T0; 1U/. The set H σ .X;Y / is clearly σ .X; Y /-compact but we will prove that aco H is not σ .X; Y /- compact, and therefore .X; µ.X; Y // cannot be quasi-complete. Indeed, proceeding σ .X;Y / by contradiction, let us assume that W VD aco H is σ .X; Y /-compact. We 0 write M.T0; 1U/ D .C.T0; 1U/; k·k1/ to denote the space of Radon measures in T0; 1U endowed with its variation norm. The map φ V X ! M.T0; 1U/ P ∗ given by φ((ξx /x2T0;1U/ D x2T0;1U ξx δx is σ .X; Y /-w -continuous. We notice that: (1) φ.W/ ⊂ φ.`1.T0; 1U//; (2) φ.W/ is an absolutely convex w∗-compact subset of M.T0; 1U/; (3) fδx V x 2 T0; 1Ug ⊂ φ.W/. From the above we obtain that ∗ w 1 BM.T0;1U/ D acofδx V x 2 T0; 1Ug ⊂ φ.W/ ⊂ φ.` .T0; 1U//; which is a contradiction because there are Radon measures on T0; 1U which are not of P the form x2T0;1U ξx δx . The proof is complete. 2 PROPOSITION 4. If X is a Banach space containing an isomorphic copy of `1.T0; 1U/, then there is a subspace Y ⊂ X 0 norm closed and norming such that .X; µ.X; Y // is not quasi-complete. PROOF. In the proof of T3, Proposition 3U the authors construct a norming subspace σ .X;E/ E ⊂ X 0 and H ⊂ X norm bounded σ .X; E/-compact such that aco H is not σ .X; E/-compact. If we take Y D E ⊂ X 0, norm closure, then norm bounded σ .X; E/-convergent nets in X are σ .X; Y /-convergent; hence we obtain that: (i) H ⊂ X is σ .X; Y /-compact; and σ .X;E/ σ .X;Y / (ii) aco H D aco H . Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.76, on 02 Oct 2021 at 11:54:59, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972709001154 412 J. Bonet and B. Cascales [4] σ .X;Y / Consequently H is σ .X; Y /-compact and aco H is not.
Load more

Recommended publications
  • Locally Convex Spaces and Schur Type Properties
    Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 44, 2019, 363–378 LOCALLY CONVEX SPACES AND SCHUR TYPE PROPERTIES Saak Gabriyelyan Ben-Gurion University of the Negev, Department of Mathematics Beer-Sheva, P.O. 653, Israel; [email protected] Abstract. In the main result of the paper we extend Rosenthal’s characterization of Banach spaces with the Schur property by showing that for a quasi-complete locally convex space E whose separable bounded sets are metrizable the following conditions are equivalent: (1) E has the Schur property, (2) E and Ew have the same sequentially compact sets, where Ew is the space E with the weak topology, (3) E and Ew have the same compact sets, (4) E and Ew have the same countably compact sets, (5) E and Ew have the same pseudocompact sets, (6) E and Ew have the same functionally bounded sets, (7) every bounded non-precompact sequence in E has a subsequence which is equivalent to the unit basis of ℓ1 and (8) every bounded non-precompact sequence in E has a subsequence which is discrete and C-embedded in Ew. 1. Introduction A locally convex space (lcs for short) E is said to have the Schur property if E and Ew have the same convergent sequences, where Ew is the space E endowed with the ′ weak topology σ(E, E ). By the classical Schur theorem the Banach space ℓ1(Γ) has the Schur property for every set Γ. The results of Rosenthal in [38] (see also §2 of [8]) show that a Banach space E has the Schur property if and only if every δ-separated sequence in the closed unit ball of E has a subsequence equivalent to the unit basis of ℓ1.
    [Show full text]
  • Mackey Continuity of Characteristic Functionals
    Georgian Mathematical Journal Volume 9 (2002), Number 1, 83–112 MACKEY CONTINUITY OF CHARACTERISTIC FUNCTIONALS S. KWAPIEN´ AND V. TARIELADZE Dedicated to N. Vakhania on the occassion of his 70th birthday Abstract. Problems of the Mackey-continuity of characteristic functionals and the localization of linear kernels of Radon probability measures in locally convex spaces are investigated. First the class of spaces is described, for which the continuity takes place. Then it is shown that in a non-complete sigma- compact inner product space, as well as in a non-complete sigma-compact metizable nuclear space, there may exist a Radon probability measure having a non-continuous characteristic functional in the Mackey topology and a linear kernel not contained in the initial space. Similar problems for moment forms and higher order kernels are also touched upon. Finally, a new proof of the result due to Chr. Borell is given, which asserts that any Gaussian Radon measure on an arbitrary Hausdorff locally convex space has the Mackey- continuous characteristic functional. 2000 Mathematics Subject Classification: Primary: 60B11. Secondary: 28C20, 60B15. Key words and phrases: Radon probability measure, characteristic func- tional, kernel of a measure, weak topology, Mackey topology, presupport, Gaussian measure, covariance operator. 1. Introduction Recall that for a Hausdorff locally convex space X the Mackey topology ¿(X¤;X) is the topology in its topological dual X¤ of uniform convergence on all weakly compact absolutely convex subsets of X. The aim of this paper is to answer the following three equivalent questions: Question 1. Is the characteristic functional¹ ˆ of a Radon probability mea- sure ¹ in a Hausdorff locally convex space X continuous in the Mackey topology ¿(X¤;X)? Question 2.
    [Show full text]
  • Mackey +0-Barrelled Spaces Stephen A
    Advances in Mathematics 145, 230238 (1999) Article ID aima.1998.1815, available online at http:ÂÂwww.idealibrary.com on Mackey +0-Barrelled Spaces Stephen A. Saxon* Department of Mathematics, University of Florida, P.O. Box 118105, Gainesville, Florida 32611-8105 E-mail: saxonÄmath.ufl.edu and Ian Tweddle* Department of Mathematics, University of Strathclyde, Glasgow G11XH, Scotland, United Kingdom E-mail: i.tweddleÄstrath.ac.uk CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - PublisherReceived Connector March 25, 1998; accepted December 14, 1998 In the context of ``Reinventing weak barrelledness,'' the best possible versions of the RobertsonSaxonRobertson Splitting Theorem and the SaxonTweddle Fit and Flat Components Theorem are obtained by weakening the hypothesis from ``barrelled'' to ``Mackey and +0-barreled.'' An example showing that the latter does not imply the former validates novelty, answers an old question, and completes a robust linear picture of ``Mackey weak barrelledness'' begun several decades ago. 1999 Academic Press Key Words: weak barrelledness; Mackey topology; splitting theorem. 1. INTRODUCTION Barrelled spaces have occupied an important place in the theory of locally convex spaces since its earliest days. This is probably due to the fact that they provide a vehicle for generalizing certain important properties of Banach spaces; for example, they form the class of domain spaces for a natural generalization of Banach's closed graph theorem and, in their dual characterization, they embody the conclusion of the BanachSteinhaus theorem for continuous linear functionals. Unlike Banach spaces, they are closed under the taking of inductive limits, countable-codimensional subspaces, products, etc. A barrel in a locally convex space is a closed * We are grateful for support and hospitality from EPSRC GRÂL67257 and Strathclyde University, and for helpful discussions with Professors J.
    [Show full text]
  • A Generalisation of Mackey's Theorem and the Uniform Boundedness Principle
    BULL. AUSTRAL. MATH. SOC. 46AO5 VOL. 40 (1989) [123-128] t A GENERALISATION OF MACKEY'S THEOREM AND THE UNIFORM BOUNDEDNESS PRINCIPLE CHARLES SWARTZ We construct a locally convex topology which is stronger than the Mackey topology but still has the same bounded sets as the Mackey topology. We use this topology to give a locally convex version of the Uniform Boundedness Principle which is valid without any completeness or barrelledness assumptions. A well known theorem of Mackey ([2, 20.11.7]) states that a subset of a locally convex topological vector space is bounded if and only if it is weakly bounded. In particular, this means that if two vector spaces X and X' are in duality, then the weak topology and the Mackey topology of X have the same bounded sets. In this note we construct a locally convex topology on X which is always stronger than the Mackey topology but which still has the same bounded sets as the Mackey (or the weak) topology. We give an example which shows that this topology which always has the same bounded sets as the Mackey topology can be strictly stronger than the Mackey topology and, therefore, can fail to be compatible with the duality between X and X'. As an application of this result, we give several versions of the Uniform Boundedness Principle for locally convex spaces which involve no completeness or barrelledness assumptions. For the remainder of this note, let X and X' be two vector spaces in duality with respect to the bilinear pairing <, > . The weak (Mackey, strong) topology on X will be denoted by cr(X, X')(T(X, X'), {3(X, X')).
    [Show full text]
  • Weak Integrals and Bounded Operators in Topological Vector Spaces
    Advances in Pure Mathematics, 2013, 3, 475-478 http://dx.doi.org/10.4236/apm.2013.35068 Published Online August 2013 (http://www.scirp.org/journal/apm) Weak Integrals and Bounded Operators in Topological Vector Spaces Lakhdar Meziani, Saud M. Alsulami Mathematics Department, King Abdulaziz University, Jeddah, KSA Email: [email protected], [email protected] Received May 4, 2013; revised June 8, 2013; accepted July 11, 2013 Copyright © 2013 Lakhdar Meziani, Saud M. Alsulami. This is an open access article distributed under the Creative Commons At- tribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is prop- erly cited. ABSTRACT Let X be a topological vector space and let S be a locally compact space. Let us consider the function space CSX0 , of all continuous functions f : SX , vanishing outside a compact set of S, equipped with an appropriate topology. In this work we will be concerned with the relationship between bounded operators TC:,0 SX X, and X-valued in- tegrals on CSX0 , . When X is a Banach space, such relation has been completely achieved via Bochner integral in [1]. In this paper we investigate the context of locally convex spaces and we will focus attention on weak integrals, namely the Pettis integrals. Some results in this direction have been obtained, under some special conditions on the structure of X and its topological dual X * . In this work we consider the case of a semi reflexive locally convex space and prove that each Pettis integral with respect to a signed measure , on S gives rise to a unique bounded operator TC:,0 SXX , which has the given Pettis integral form.
    [Show full text]
  • The Mackey Topology As a Mixed Topology 109
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 53, Number I, November 1975 THE MACKEYTOPOLOGY AS A MIXEDTOPOLOGY J. B. COOPER ABSTRACT. A theorem on the coincidence of a mixed topology and the Mackey topology is given. The techniques of partitions of unity are used. As a corollary, a result of LeCam and Conway on the strict topology is obtained. Similar methods are applied to obtain a theorem of Collins and Dorroh. Introduction. In recent years the space C(S) of bounded, continuous complex-valued functions on a locally compact space S with the strict topol- ogy has received a great deal of attention. In [5] we have shown that this to- pology was a special case of a mixed topology in the sense of the Polish school. Many of the results on the strict topology are, in fact, special cases of general results on mixed topologies and, when this is the case, it seems to us to be desirable to present this approach since it displays the strict to- pology as part of a larger scheme rather than as an isolated phenomenon. In this note, we give an example of this process by considering the prob- lem: When is the strict topology on C(S) the Mackey topology? LeCam and Conway have shown that this is the case when S is paracompact. In fact, there are a number of results of this type, that is, identifying mixed topologies with the Mackey topology (see Cooper [6], Stroyan Ll3]) and this suggests that there might be a general theorem of this type in the context of mixed to- pologies.
    [Show full text]
  • The Strict Topology and Compactness in the Space of Measures
    Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1965 The trS ict Topology and Compactness in the Space of Measures. John Bligh Conway Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses Recommended Citation Conway, John Bligh, "The trS ict Topology and Compactness in the Space of Measures." (1965). LSU Historical Dissertations and Theses. 1068. https://digitalcommons.lsu.edu/gradschool_disstheses/1068 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact [email protected]. This dissertation has been microfilmed exactly as received 6 6—7 24 CONWAY, John Bligh, 1939- THE STRICT TOPOLOGY AND COMPACTNESS IN THE SPACE OF MEASURES. Louisiana State University, Ph.D., 1965 Mathematics University Microfilms, Inc., Ann Arbor, Michigan Reproduced with permission of the copyright owner. Further reproduction prohibitedpermission. without THE STRICT TOPOLOGY AND COMPACTNESS IN THE SPACE OF MEASURES A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Mathematics John B/^Conway B.S., Loyola University, 196I August, 1965 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENT The author wishes to express his appreciation to Professor Heron S. Collins for his advice and encourage­ ment. This dissertation was written while the author held a National Science Foundation Cooperative Fellowship.
    [Show full text]
  • Arxiv:2005.05202V4 [Math.FA]
    BOUNDED TOPOLOGIES ON BANACH SPACES AND SOME OF THEIR USES IN ECONOMIC THEORY: A REVIEW ANDREW J. WROBEL Abstract. Known results are reviewed about the bounded and the convex bounded variants, bT and cbT , of a topology T on a real Banach space. The focus is on the cases of T = w (P ∗, P ) and of T = m (P ∗, P ), which are the weak* and the Mackey topologies on a dual Banach space P ∗. The convex bounded Mackey topology, cbm (P ∗, P ), is known to be identical to m (P ∗, P ). As for bm (P ∗, P ), it is conjectured to be strictly stronger than m (P ∗, P ) or, equivalently, not to be a vector topology (except when P is reflexive). Some uses of the bounded Mackey and the bounded weak* topologies in economic theory and its applications are pointed to. Also reviewed are the bounded weak and the compact weak topologies, bw(Y, Y ∗) and kw(Y, Y ∗), on a general Banach space Y , as well as their convex variants (cbw and ckw). 1. Introduction Nonmetric topologies on the norm-dual, P ∗, of a real Banach space (P ) can become much more manageable when restricted to bounded sets. For example, given a convex subset of P ∗, or a real-valued concave function on P ∗, the bounded weak* topology, ∗ bw := bw (P ∗,P ), can serve to show that the set in question is weakly* closed, or that the function is weakly* upper semicontinuous. In economic theory, such uses of the Krein-Smulian Theorem are made in [9, Proposition 1.1, Theorems 4.4 and 4.7], [22, Proposition 1 and Example 5], [24, Lemma 4.1] and [25, Section 6.2].
    [Show full text]
  • A Strong Uniform Boundedness Result on $\Kappa
    Kangweon-Kyungki Math. Jour. 4 (1996), No. 1, pp. 1–5 A STRONG UNIFORM BOUNDEDNESS RESULT ON κ-SPACES Min-Hyung Cho Abstract. A strong Banach-Mackey property is established for κ- spaces including all complete and some non-complete metric linear spaces and some non-metrizable locally convex spaces. As applica- tions of this result, a strong uniform boundedness result and a new Banach-Steinhaus type theorem are obtained. A topological vector space X is said to be a κ-space if every null sequence {xn} in X has a subsequence {xnk } such that the series P∞ k=1 xnk converges in X ([1], [2]). κ-spaces make a large class con- taining complete metric linear spaces, some non-complete metric lin- ear spaces and some non-metrizable locally convex spaces. Recently, κ-spaces have been shown to enjoy many nice properties ([1],[2],[3],[4]). In this paper, we would like to establish a strong Banach-Mackey prop- erty for κ-spaces and derive from this several important results includ- ing a strong uniform boundedness principle and a Banach-Steinhaus type theorem. For the remainder of this note, (X, τ) represents a topological vector space with the vector space X and the vector topology τ on X. Some- times, we use only X instead of (X, τ). For a topological vector space X let X0, Xs and Xb denote the families of continuous, sequentially continuous and bounded linear functionals on X, respectively ([5]). Similarly, for topological vector spaces X and Y let L(X, Y ), SC(X, Y ) and B(X, Y ) denote the families of continuous, sequentially continuous and bounded linear operators from X into Y , respectively.
    [Show full text]
  • Topological Characterization of Weakly Compact Operators Revisited †
    E extracta mathematicae Vol. 22, N´um. 2, 215 – 223 (2007) Satellite conference on Banach spaces · ICM2006 Topological characterization of weakly compact operators revisited † Antonio M. Peralta Departamento de An´alisisMatem´atico, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain, [email protected] Received December 1, 2006 Abstract: In this note we revise and survey some recent results established in [8]. We shall show that for each Banach space X, there exists a locally convex topology for X, termed the “Right Topology”, such that a linear map T, from X into a Banach space Y, is weakly compact, precisely when T is a continuous map from X, equipped with the “Right” topology, into Y equipped with the norm topology. We provide here a new and shorter proof of this result. We shall also survey the results concerning sequentially Right-to-norm continuous operators. Key words: Weakly compact operator, Right topology, Mackey topology, Property (V) AMS Subject Class. (2000): 47B07, 47A20 1. Introduction This note is thought to be a complement of the talk presented at the conference “Banach Space Theory, C´aceres 2006”. In the lecture given at the conference, the results treated in [8] were presented only tangentially. In this note we shall survey the topological characterization of weakly compact operators obtained in [8]. On the other hand, in [8] we strived for maximal clarity rather than maximal generality or maximal conciseness. Among the novelties included in this note, we present an alternative and shorter proof for this topological characterization of weakly compact operators. In the last part of section §2, we shall review the connections between the Right topology and some previous studies, on certain locally convex topologies associated with operator ideals, due to I.
    [Show full text]
  • A Characterization of Separable Conjugate Spaces
    A CHARACTERIZATION OF SEPARABLE CONJUGATE BANACH SPACES By STEFANO ROSSI Abstract The following elegant characterization of dual Banach spaces will be proved: a separable Banach space X is a dual space if and only exists a norm closed subspace M ⊂ X∗ with (Dixmier) characteristic equal to 1, whose functionals are norm-attaining. Contents 1 Introduction 1 2 Notations and preliminaries 2 3 A general result 2 4 The separable case 4 5 Applications 7 1 Introduction Among all Banach spaces, dual spaces have some additional properties, which assure a more far-reaching treatment of their structure. Probably the most useful property is represented by weak*-compactness of the unit ball of such spaces. Giving necessary and sufficient conditions for a Banach space X to be a dual space is a long standing problem; it goes back essentially to J. Dixmier [3]. Many theorems concerning this intriguing area are known; these results usually require the compactness property for X1, the unit ball of X, with re- spect to suitable locally convex topology. Elegant as theoretical results, these arXiv:1003.2224v1 [math.FA] 10 Mar 2010 theorems, however, do not provide any practical tool to solve the question in the applications, because compactness property is very difficult to be checked in infinite-dimensional contexts. In spite of this, reflexive Banach spaces (which are indeed dual spaces with unique predual) are well understood thanks to a celebrate thorem by R. C. James [4], which states that a Banach space X is reflexive if and only if every continuous linear functional attains its norm on X1.
    [Show full text]
  • Some Remarks on the Mackey Topology in a Banach Space
    The Mackey topology and coarser Mackey topologies The dual Mackey topology Some remarks on the Mackey topology in a Banach space A. J. Guirao1, V. Montesinos1 1Instituto de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Spain IX Encuentro de la Red de Análisis Funcional. Zafra, 2013 A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space The Mackey topology and coarser Mackey topologies The dual Mackey topology Outline 1 The Mackey topology and coarser Mackey topologies 2 The dual Mackey topology A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space w(E; F) the weak topology on E associated to hE; Fi. w(E; F) is the coarsest lcs topology on E compatible with hE; Fi. K all the absolutely convex w(F; E)-compact subsets of F. µ(E; F) := TK, the Mackey topology on E. Theorem (Mackey–Arens) µ(E; F) is the finest lcs topology on E compatible with hE; Fi. The Mackey topology and coarser Mackey topologies The dual Mackey topology The Mackey topology associated to a dual pair hE; Fi a dual pair. A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space w(E; F) is the coarsest lcs topology on E compatible with hE; Fi. K all the absolutely convex w(F; E)-compact subsets of F. µ(E; F) := TK, the Mackey topology on E. Theorem (Mackey–Arens) µ(E; F) is the finest lcs topology on E compatible with hE; Fi. The Mackey topology and coarser Mackey topologies The dual Mackey topology The Mackey topology associated to a dual pair hE; Fi a dual pair.
    [Show full text]