DOCSLIB.ORG

Normed Spaces Which Are Not Mackey Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Normed Spaces Which Are Not Mackey Groups axioms Article Normed Spaces Which Are Not Mackey Groups Saak Gabriyelyan Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 8410501, Israel; [email protected] Abstract: It is well known that every normed (even quasibarrelled) space is a Mackey space. However, in the more general realm of locally quasi-convex abelian groups an analogous result does not hold. We give the first examples of normed spaces which are not Mackey groups. Keywords: normed space; Mackey group; locally quasi-convex; compatible group topology MSC: 46A8; 46E10; 54H11 1. Introduction Let (E, t) be a locally convex space (lcs for short). A locally convex vector topology n on E is called compatible with t if the spaces (E, t) and (E, n) have the same topological dual space. The famous Mackey–Arens Theorem states that there is a finest locally convex vector space topology m on E compatible with t. The topology m is called the Mackey topology on E = E Mackey space associated with t, and if m t, the space is called a . The most important class of Mackey spaces is the class of quasibarrelled spaces. This class is sufficiently rich and contains all metrizable locally convex spaces. In particular, every normed space is a Citation: Gabriyelyan, S. Normed Mackey space. Spaces Which Are Not Mackey ( ) Groups. Axioms 2021, 10, 217. For an abelian topological group G, t we denote by Gb the group of all continuous https://doi.org/10.3390/ characters of (G, t). Two topologies m and n on an abelian group G are said to be compatible axioms10030217 if (\G, m) = (\G, n). Being motivated by the concept of Mackey spaces, the following notion was implicitly introduced and studied in [1], and explicitly defined in [2] (for all relevant Academic Editor: Feliz Manuel definitions see the next section): A locally quasi-convex abelian group (G, m) is called a Minhós Mackey group if for every locally quasi-convex group topology n on G compatible with m it follows that n ≤ m. Received: 14 July 2021 Every lcs considered as an abelian topological group is locally quasi-convex. So, it Accepted: 1 September 2021 is natural to ask whether every Mackey space is also a Mackey group. Surprisingly, the Published: 8 September 2021 answer to this question is negative. Indeed, answering a question posed in [2], we show in [3] that there is even a metrizable lcs which is not a Mackey group. Recall that for every Publisher’s Note: MDPI stays neutral Tychonoff space X, the space Cp(X) of all continuous functions on X endowed with the with regard to jurisdictional claims in pointwise topology is quasibarrelled, and hence it is a Mackey space. However, in [4] we published maps and institutional affil- proved that the space Cp(X) is a Mackey group if and only if it is barrelled. In particular, iations. the metrizable space Cp(Q) is not a Mackey group. These results motivate the following ≤ ≤ T (N) = S n question. For 1 p ¥, denote with `p the topology on the direct sum R : n2N R induced from `p. Copyright: © 2021 by the author. Problem 1 ([3]). Does there exist a normed space E which is not a Mackey group? What about Licensee MDPI, Basel, Switzerland. (N) (R , T`p )? This article is an open access article distributed under the terms and The main goal of this note is to answer Problem1 in the affirmative. More pre- conditions of the Creative Commons (N) 1 (N) cisely, we show that the normed spaces c00 := ( , T` ) and ` := ( , T` ) are not Attribution (CC BY) license (https:// R ¥ 00 R 1 Mackey groups. creativecommons.org/licenses/by/ 4.0/). Axioms 2021, 10, 217. https://doi.org/10.3390/axioms10030217 https://www.mdpi.com/journal/axioms Axioms 2021, 10, 217 2 of 6 2. Main Result Set N := f1, 2, ... g. Denote by S the unit circle group and set S+ := fz 2 S : Re(z) ≥ 0g. Let G be an abelian topological group. A character c 2 Gb is a continuous homomor- phism from G into S. A subset A of G is called quasi-convex if for every g 2 G n A there exists c 2 Gb such that c(g) 2/ S+ and c(A) ⊆ S+. An abelian topological group is called locally quasi-convex if it admits a neighborhood base at the neutral element 0 consisting of quasi-convex sets. It is well known that the class of locally quasi-convex abelian groups is closed under taking products and subgroups. The following group plays an essential role in the proof of our main results, Theorems1 and2. Set N c0(S) := (zn) 2 S : zn ! 1 , 1 2 1 and denote by F0(S) the group c0(S) endowed with the metric d (zn), (zn) = supfjzn − 2 N znj, n 2 Ng. Then F0(S) is a Polish group, and the sets of the form V \ c0(S), where V is a neighborhood at the unit 1 2 S, form a base at the identity 1 = (1n) 2 F0(S). In [5] (Theorem1), we proved that the group F0(S) is reflexive and hence locally quasi-convex. A proof of the next important result can be found in [6] [Proposition 2.3]. Fact 1. Let E be a real lcs. Then the map y : E0 ! Eb, y(c) := e2pic, is an algebraic isomorphism. We use the next standard notations. Let fengn2N be the standard basis of the Banach ∗ 0 space (c0, k · k¥), and let fengn2N be the canonical basis in the dual Banach space (c0) = `1, i.e., ∗ en = (0, . , 0, 1, 0, . ) and en = (0, . , 0, 1, 0, . ), (N) T where 1 is placed in position n. Then c00 = (R , `¥ ) is a dense subspace of c0 consisting of all vectors with finite support. Theorem 1. The normed space c00 is not a Mackey group. T Proof. For simplicity and clearness of notations we set E := c00 and t := `¥ . For every ∗ ∗ 0 n 2 N, set cn := nen. It is clear that cn ! 0 in the weak topology on E and hence in s(Eb, E). Therefore we can define the linear injective operator F : E ! E × c0 and the monomorphism p : E ! E × F0(S) setting (for all x = (xn) 2 E) F(x) := x, R(x) , where R(x) := cn(x) = (nxn) 2 c0, p(x) := x, R0(x) , where R0(x) := Q ◦ R(x) = expf2picn(x)g = expf2pinxng 2 F0(S). Denote with T and T0 the topologies on E induced from E × c0 and E × F0(S), re- spectively. So T is a locally convex vector topology on E and T0 is a locally quasi-convex group topology on E. By construction, t ≤ T0 ≤ T, so taking into account Fact1 and the Hahn–Banach extension theorem, we obtain 0 0 y(E ) = y `1 ⊆ \E, T0 ⊆ y (E, T) ⊆ y `1 × `1 . (1) Step 1: The topologies t and T0 are compatible. By (1), it is sufficient to show that each continuous character of E, T0 belongs to y `1 . Fix c 2 \E, T0 . Then (1) implies that c = y(h) = expf2pihg for some h = n, (cn) 2 `1 × `1, where n 2 `1 and (cn) 2 `1, and h(x) = n(x) + ∑ cncn(x) = n(x) + ∑ cn · nxn x = (xn) 2 E . n2N n2N 2 ` 2 ` To prove that c y 1 it is sufficient (and also necessary) to show that cnn n 1. Replacing, if needed, h by h − n, we assume that n = 0. Axioms 2021, 10, 217 3 of 6 Suppose for a contradiction that ∑n jcnjn = ¥. Since c is continuous, Fact1 shows that, for every # < 0.01, there is a d < # such that h(x) = ∑ ncnxn 2 (−#, #) + Z, for every x 2 Ud, (2) n2N where Ud is a canonical T0-neighborhood of zero Ud := x = (xn) 2 E : jxnj ≤ d and nxn 2 [−d, d] + Z for every n 2 N . (3) In what follows # and d are fixed as above. We distinguish between three cases. f g ⊆ j j ! ! Case 1: There is a subsequence nk k2N N such that cnk nk ¥ as k ¥. As jcnk jnk ! ¥ and cn ! 0, there is k 2 N such that 1 > 3 < jc j 1 and jc jn d. (4) 8 nk 8 nk k The first inequality in (4) implies that there is 2 1 3 \ mk jc j , jc j N. (5) 8 nk 8 nk mk Set x = (xn) := (0, ... , 0, sign(cn ) , 0, ... ), where the nonzero element is placed in k nk position nk. Then nxn 2 Z for every n 2 N, and the second inequality of (4) and (5) imply k k = j j = mk < 3 < x ¥ xnk n jc jn d. k 8 nk k Therefore x 2 Ud. On the other hand, (5) implies 1 mk 3 < h(x) = cnnxn = jcn jn = jcn jm < . 8 ∑ k k nk k k 8 n2N Hence h(x) 62 (−#, #) + Z since # < 0.01. However, this contradicts (2). f g > j j ! Case 2: There is a subsequence nk k2N and a number a 0 such that cnk nk a as k ! ¥. Choose N 2 N such that a < j j < 3a ≥ 2 cnk nk 2 for all k N. (6) Choose a finite subset F of fnkgk≥N and, for every n 2 F, a natural number in such that the following two conditions are satisfied: in 2 f1, 2, . , bdncg for every n 2 F, (7) and 10 in 30 72a < ∑ n < 72a (8) n2F 1 in bdnc 10 (this is possible because n ≤ n ≤ n ≈ d and n ! ¥: so, if 72a < d the set F can be 10 chosen to have only one element, and if d ≤ 72a , the set F also can be easily chosen to be finite).
Load more

Recommended publications
  • Subspaces and Quotients of Topological and Ordered Vector Spaces
    Zoran Kadelburg Stojan Radenovi´c SUBSPACES AND QUOTIENTS OF TOPOLOGICAL AND ORDERED VECTOR SPACES Novi Sad, 1997. CONTENTS INTRODUCTION::::::::::::::::::::::::::::::::::::::::::::::::::::: 1 I: TOPOLOGICAL VECTOR SPACES::::::::::::::::::::::::::::::: 3 1.1. Some properties of subsets of vector spaces ::::::::::::::::::::::: 3 1.2. Topological vector spaces::::::::::::::::::::::::::::::::::::::::: 6 1.3. Locally convex spaces :::::::::::::::::::::::::::::::::::::::::::: 12 1.4. Inductive and projective topologies ::::::::::::::::::::::::::::::: 15 1.5. Topologies of uniform convergence. The Banach-Steinhaus theorem 21 1.6. Duality theory ::::::::::::::::::::::::::::::::::::::::::::::::::: 28 II: SUBSPACES AND QUOTIENTS OF TOPOLOGICAL VECTOR SPACES ::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 39 2.1. Subspaces of lcs’s belonging to the basic classes ::::::::::::::::::: 39 2.2. Subspaces of lcs’s from some other classes :::::::::::::::::::::::: 47 2.3. Subspaces of topological vector spaces :::::::::::::::::::::::::::: 56 2.4. Three-space-problem for topological vector spaces::::::::::::::::: 60 2.5. Three-space-problem in Fr´echet spaces:::::::::::::::::::::::::::: 65 III: ORDERED TOPOLOGICAL VECTOR SPACES :::::::::::::::: 72 3.1. Basics of the theory of Riesz spaces::::::::::::::::::::::::::::::: 72 3.2. Topological vector Riesz spaces ::::::::::::::::::::::::::::::::::: 79 3.3. The basic classes of locally convex Riesz spaces ::::::::::::::::::: 82 3.4. l-ideals of topological vector Riesz spaces :::::::::::::::::::::::::
    [Show full text]
  • COUNTABLE-CODIMENSIONAL SUBSPACES of LOCALLY CONVEX SPACES by J
    COUNTABLE-CODIMENSIONAL SUBSPACES OF LOCALLY CONVEX SPACES by J. H. WEBB (Received 9th December 1971) A barrel in a locally convex Hausdorff space E[x] is a closed absolutely convex absorbent set. A a-barrel is a barrel which is expressible as a countable intersection of closed absolutely convex neighbourhoods. A space is said to be barrelled {countably barrelled) if every barrel (a-barrel) is a neighbourhood, and quasi-barrelled (countably quasi-barrelled) if every bornivorous barrel (ff-barrel) is a neighbourhood. The study of countably barrelled and countably quasi-barrelled spaces was initiated by Husain (2). It has recently been shown that a subspace of countable codimension of a barrelled space is barrelled ((4), (6)), and that a subspace of finite codimension of a quasi-barrelled space is quasi-barrelled (5). It is the object of this paper to show how these results may be extended to countably barrelled and countably quasi-barrelled spaces. It is known that these properties are not preserved under passage to arbitrary closed subspaces (3). Theorem 6 shows that a subspace of countable codimension of a countably barrelled space is countably barrelled. Let {£„} be an expanding sequence of subspaces of E whose union is E. 00 Then £' s f] E'n, and the reverse inclusion holds as well (1) if either (i) E'\o(E', E)~\ is sequentially complete or (ii) E'[fi(E', £)] is sequentially complete, and every bounded subset of E is contained in some En. Theorem 1. Let E[%\ be a locally convex space with r = n(E, E') (the Mackey CO topology).
    [Show full text]
  • Tightness and Distinguished Fréchet Spaces ✩
    J. Math. Anal. Appl. 324 (2006) 862–881 www.elsevier.com/locate/jmaa Tightness and distinguished Fréchet spaces ✩ J.C. Ferrando a,J.K˛akol b, M. López Pellicer c,S.A.Saxond,∗ a Centro de Investigación Operativa, Universidad M. Hernández, E-03202 Elche (Alicante), Spain b Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614 Pozna´n, Poland c Departamento de Matemática Aplicada y IMPA, Universidad Politécnica, E-46022 Valencia, Spain d Department of Mathematics, University of Florida, PO Box 11805, Gainesville, FL 32611-8105, USA Received 22 April 2005 Available online 20 January 2006 Submitted by T.D. Benavides Abstract Valdivia invented a nondistinguished Fréchet space whose weak bidual is quasi-Suslin but not K-analytic. We prove that Grothendieck/Köthe’s original nondistinguished Fréchet space serves the same purpose. In- deed, a Fréchet space is distinguished if and only if its strong dual has countable tightness, a corollary to the fact that a (DF)-space is quasibarrelled if and only if its tightness is countable. This answers a Cas- cales/Kakol/Saxon ˛ question and leads to a rich supply of (DF)-spaces whose weak duals are quasi-Suslin but not K-analytic, including the spaces Cc(κ) for κ a cardinal of uncountable cofinality. Our level of gen- erality rises above (DF)- or even dual metric spaces to Cascales/Orihuela’s class G. The small cardinals b and d invite a novel analysis of the Grothendieck/Köthe example, and are useful throughout. © 2006 Elsevier Inc. All rights reserved. Keywords: Class G; Quasibarrelled; K-analytic; Quasi-Suslin; Compact-open ✩ The research of the second author was supported by the Komitet Badan´ Naukowych (State Committee for Scientific Research), Poland, grant no.
    [Show full text]
  • 2-Symmetric Locally Convex Spaces
    2-SYMMETRIC LOCALLYCONVEX SPACES D. E. EDMUNDS In [l] it is shown that barrelledness and quasi-barrelledness are merely the two extreme examples of a property, called 2-symmetry, which may be possessed by a locally convex Hausdorff topological vector space. The object of this note is to show how recent char- acterisations [2; 3] of barrelled and quasi-barrelled spaces may be subsumed under characterisations of 2-symmetric spaces, and to ex- hibit some properties of these spaces. First we need some definitions and simple results. 1. Let £ be a locally convex Hausdorff topological vector space (abbreviated to LCS in what follows), and let S be a class of bounded subsets of E whose union is E. Let E' denote the topological dual of E, and let E'z be the set E' endowed with the topological of uniform convergence on the members of 2. Definition 1. A subset of E is said to be X-bornivorous if it absorbs every member of 2. Definition 2. We say that E is "2-symmetric if any of the following equivalent conditions hold : (a) Every S-bornivorous barrel in £ is a neighbourhood of zero. (b) Every bounded subset of E'j¡ is equicontinuous. (c) The topology induced on E by the strong dual of E% is the original topology of E. The equivalence of these conditions was proved in [l]. If 2iG22 it is easy to see that Si-symmetry implies 22-symmetry ; the strongest restriction on E is obtained by taking for 2 the class s of all subsets of E consisting of a single point, and then S-symmetry is simply the property of being barrelled.
    [Show full text]
  • 181. on Nuclear Spaces with Fundamental System O F Bounded Sets
    No. 8] Proc. Japan Acad., 44 (1968) 807 181. On Nuclear Spaces with Fundamental System o f Bounded Sets. II By Shunsuke FUNAKOs1 (Comm. by Kinjiro KuNUGI, M. J. A., Oct. 12, 1968) A locally convex vector space with a countable fundamental system of bounded sets has already been developed in several bibliog- raphies. Barrelled spaces and quasi-barrelled spaces with a count- able fundamental system of compact sets has been studied by J. Dieudonne [2] and by M. Mahowald and G. Gould [7] respectively. We considered, the open mapring and closed graph theorems on a nuclear dualmetric space in the previous paper [4]. Let E be a nonmed space then E is a nuclear space if and only if it is finite dimentional. It is also known that a nonmed space can only be a Montel (i.e., barrelled and perfect) space if it is finite dimen- sional. In this paper, we prove a nuclear dualmetric space which is quasi-complete is Montel space, and using this result, we consider analogous theorem to M. Mahowald and G. Gould [7], in nuclear space. For nuclear spaces and its related notion, see A. Pietsch [8] and S. Funakosi [4]. Most of the definitions and notations of the locally convex vector spaces are taken' from N. Bourbaki [1] and T. Husain [5]. Definition. Let E be a locally convex space and E' its dual. (1) I f only all countable strong bounded subset o f E' are equi- continuous, then E is called the oW-quasi-barrelled. (2) Let E be a a-quasi-barrelled space, if there exists a countable fundamental system o f bounded subset in E, then E is called the dual- metric space.
    [Show full text]
  • Stojan Radenovi C Zoran Kadelburg
    Scientiae Mathematicae Vol.1, No. 11998, 43{49 43 ON A. GROTHENDIECK'S PROBLEMS CONCERNING F AND DF{SPACES Stojan Radenovic Zoran Kadelburg For non-exp erts only Received Novemb er 7, 1996; revised March 6, 1997 Abstract. This is an exp ository article devoted to the answers to the ten op en problems from the fundamental pap er [16] of A. Grothendieck, some of whichwere given in the recent years. A signi cant contribution to the development of the theory of lo cally convex spaces lcs , as a part of functional analysis, was given in the fties by a group of French mathematicians L. Schwartz, N. Bourbaki, J. Dieudonn e, A. Grothendieck. The dominant part in their work, concerning this part of functional analysis, was devoted to the so called duality theory.Itwas esp ecially applied to the investigation of metrizable, i.e. Fr echet lcs, which are \nearest" to the normed and Banach spaces [14]. The article [16] of A. Grothendieck \on F and DF spaces" takes a sp ecial place in the theory of lcs. Apart from solving nearly all the problems of L. Schwartz and J. Dieudonn e, p osed in [14], a signi cant class of spaces|\DF-spaces"|was intro duced in it. This class contains a lot of imp ortant functional spaces|we mention just a few, following the pap er [16] itself: m 0 0 m 0 a The spaces E , E , D , D , intro duced byL.Schwartz in [28] for ap- L L p p plication in the theory of distributions; also the space C of continuous functions with compact supp ort.
    [Show full text]
  • Abdus Salam United Nations Educational, Scientific and Cultural International Organization Centre
    the Miii abdus salam united nations educational, scientific and cultural international organization centre international atomic energy agency for theoretical physics ON WEIGHTED SPACES WITHOUT I A FUNDAMENTAL SEQUENCE OF BOUNDED SETS J.O. Olaleru 4 Available at: http://www. ictp.trieste.it/~pub_ off IC/2001/7 United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS ON WEIGHTED SPACES WITHOUT A FUNDAMENTAL SEQUENCE OF BOUNDED SETS J.O. Olaleru1 Mathematics Department, University of Lagos, Yaba, Lagos and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. Abstract The problem of countably quasibarrelledness of weighted spaces of continuous functions, of which there are no results in the general setting of weighted spaces, is takcled in this paper. This leads to the study of quasibarrelledness of weighted spaces which, unlike that of Ernst and Schnettler[4], though with a similar approach, we drop the assumption that the weighted space has a fundamental sequence of bounded sets. The study of countably quasibarrelledness of weighted spaces naturally leads to definite results on the weighted (DF)-spaces for those weighted spaces with a fundamental sequence of bounded sets. MIRAMARE - TRIESTE September 2001 'E-mail: [email protected] 1. Introduction and notations The countably barrelledness,countably quasibarrelledness, barrelledness, quasibarrelleness of the space C(X) of continuous functions on a completely regular Hausdorff space X equipped with the compact open topology(c-op), including when it is a (DF)-space and (gDF)-space, is well known (e.g.see [8] and [14]). In the more general setting of weighted spaces, Ernst and Schnettler[4] studied the (gDF) and the quasibarrelledness of weighted spaces by constructing a Nachbin family on X which is based on the assumption that the weighted space has a fundamental sequence of bounded sets.
    [Show full text]
  • Kyungpook Math. J Volume 26, Number 2 on SEQUENTIAL BARRELLEDNESS of LOCALLY CONVEX SPACES by W. H. Hsiang 1. Introduction Let L
    Kyungpook Math. J Volume 26, Number 2 December、 1986 ON SEQUENTIAL BARRELLEDNESS OF LOCALLY CONVEX SPACES By W. H. Hsiang 1. Introduction Let L be a locally convex space (abbrev. by LCS) over K (K=R Or K=C) , and L' be the dual (linear) space of all continuous, linear functionals on L. L is called countably barrelled (or quasibarrelled) if ∞unded subsets of L。’ (or L/) which are countable unions of equicontinuous subsets of L' are equicontinuous on L-Defs. 1 and 2, [6 ]. Hence countably barrelled spaces are countably quasibarrelled. Conversely, Hausdorff, quasi-complete, countably quasibarrelled spaces are countably barrelled. Moreover, the completion L (Thm. 2. 9.1, [3] _) of any Hausdorff, countably quasibarrelled space L is countably barrelled. AlI the above results are Propositions 3, 4 and 5, [6]. Besides these relations between countably barrelled and countably quasibarrelled spaces, there are other interesting properties in [6] _ For example, Theorem 3 is the Banach-Steinhaus theorem, and Thenrem 6 is on the m.pping proper­ ties of countably barrelled spaces. On the other hand, the spaces mentioned in the last paragraph had been extended to other classes of barrelled spaces. For example, a LCS L is called a-convergently barrelled (or sequeη t ially barrelled in De f. vr. 4, [7].; or on p.353, [1 3].) if convergent sequences in L_' are equicontinuous on L. In this ’ case, L/ and Lß’ have the same bounded subsets; and L has the same barrels and quasibarrels (prop. 4(i) and (ii) , [13].), where a barrel (or quasibarrel) 。 f L is a
    [Show full text]
  • The Space of Pettis Integrable Functions Is Barrelled
    proceedings of the american mathematical society Volume 114, Number 3, MARCH 1992 THE SPACE OF PETTIS INTEGRABLE FUNCTIONS IS BARRELLED LECH DREWNOWSKI,MIGUEL FLORENCIO, AND PEDRO J. PAÚL (Communicated by William J. Davis) Abstract. It is well known that the normed space of Pettis integrable func- tions from a finite measure space to a Banach space is not complete in general. Here we prove that this space is always barrelled; this tells us that we may ap- ply two important results to this space, namely, the Banach-Steinhaus uniform boundedness principle and the closed graph theorem. The proof is based on a theorem stating that a quasi-barrelled space having a convenient Boolean alge- bra of projections is barrelled. We also use this theorem to give similar results for the spaces of Bochner integrable functions. 1. Barrelledness of spaces with Boolean algebras of projections This paper deals with some conditions, related to Boolean algebras of projec- tions modelled over a finite measure space, under which quasi-barrelled spaces are indeed barrelled. Barrelled spaces were introduced as the "good" class of locally convex spaces, as domain spaces for operators, for which uniform bound- edness principles and closed graph theorems hold (see [6, §39.5(1), §34.6(9)] or [10, 4.1.3, 4.1.10, 7.1.12]). Although we refer the reader to the monographs [2, 6, 10] for the terms used in this paper, we want to recall briefly the most relevant definitions. Let F be a locally convex space with dual E'. Denote, respectively, by ct(F/ , E) and /?(F', E) the corresponding weak and strong topologies of the duality on E'.
    [Show full text]
  • Fundamental Bounded Resolutions and Quasi-$(DF) $-Spaces
    FUNDAMENTAL BOUNDED RESOLUTIONS AND QUASI-(DF )-SPACES J. C. FERRANDO, S. GABRIYELYAN, AND J. KA¸KOL Abstract. We introduce a new class of locally convex spaces E, under the name quasi-(DF )- spaces, containing strictly the class of (DF )-spaces. A locally convex space E is called a quasi- (DF )-space if (i) E admits a fundamental bounded resolution, i.e. an NN-increasing family of bounded sets in E which swallows all bounded set in E, and (ii) E belongs to the class G (in sense of Cascales–Orihuela). The class of quasi-(DF )-spaces is closed under taking subspaces, countable direct sums and countable products. Every regular (LM)-space (particularly, every metrizable locally convex space) and its strong dual are quasi-(DF )-spaces. We prove that Cp(X) has a fundamental bounded resolution iff Cp(X) is a quasi-(DF )-space iff the strong dual of Cp(X) is a quasi-(DF )-space iff X is countable. If X is a metrizable space, then Ck(X) is a quasi-(DF )-space iff X is a Polish σ-compact space. We provide numerous concrete examples which in particular clarify differences between (DF )-spaces and quasi-(DF )-spaces. 1. Introduction An important class of locally convex spaces (lcs, for short), the class of (DF )-space, was intro- duced by Grothendieck in [20], we refer also to monographs [21] or [31] for more details. Definition 1.1. A locally convex space E is called a (DF )-space if (1) E has a fundamental sequence of bounded sets, and (2) E is ℵ0-quasibarrelled.
    [Show full text]
  • On Realcompact Topological Vector Spaces
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Springer - Publisher Connector RACSAM (2011) 105:39–70 DOI 10.1007/s13398-011-0003-0 SURVEY On realcompact topological vector spaces J. K¸akol · M. López-Pellicer Received: 7 January 2010 / Accepted: 7 May 2010 / Published online: 3 February 2011 © The Author(s) 2011. This article is published with open access at Springerlink.com Abstract This survey paper collects some of older and quite new concepts and results from descriptive set topology applied to study certain infinite-dimensional topological vec- tor spaces appearing in Functional Analysis, including Fréchet spaces, (LF)-spaces, and their duals, (DF)-spaces and spaces of continuous real-valued functions C(X) on a completely regular Hausdorff space X. Especially (LF)-spaces and their duals arise in many fields of Functional Analysis and its applications, for example in Distributions Theory, Differential Equations and Complex Analysis. The concept of a realcompact topological space, although originally introduced and studied in General Topology, has been also studied because of very concrete applications in Linear Functional Analysis. Keywords Angelicity · Baire and (b-) Baire-like · Bornological · Borel set · C∗-embedded · Class G · (DF) space · Distinguished space · Fréchet–Urysohn · k-Space · K -Analytic · (Weakly) Lindelöf (Σ) · Locally convex space · (Σ-)Quasi-Suslin space · (Strongly) realcompact space · (Compact) resolution · Talagrand compact · (Countable) tightness · Trans-separable · Weakly compact · (WCG) space · Web-bounded (compact) Mathematics Subject Classification (2000) 54H05 · 46A04 · 46A50 Dedicated to Professor Manuel Valdivia, excellent professor and mathematical researcher, on the occasion of his 80th birthday. The research for the first named author was (partially) supported by Ministry of Science and Higher Education, Poland, Grant no.
    [Show full text]
  • Math. Ann 189 (1970), 280-284
    BIBLIOGRAPHY [1] Adasch, N.; Topologische Produkte gewisser topologischer Vektorraume; Math. Ann 189 (1970), 280-284. [2] Arsove, M.G. and Edwards, R.E.; Generalized bases in topological linear spaces, Studia Math. 19 (1960), 95-113. [3] Banach, s. Theorie des operations lineaires, Warsaw (1932). [4] Bourbaki, N.; Espaces vectoriels topologiques, Chapters I-V Herman, Paris (1953,1955). [5] Bozel, F. and Husain, T.; On isomorphisms of locally convex spaces with similar biorthogonal systems, Canadian Math. Bull. Vol. 16 (2) (1973). [6] De Wilde, M.; sur les sous-espaces de codimension finie d'un espace lineaire a seminormes. Bull. Soc. Royale de Sci. Liege n° 9-10 (1969), 450-453. [7] De Wilde, M. and Houet, C.; On increasing sequences of absolutely convex sets in locally convex spaces, Math. Ann. 192 (1971), 257-261. [8] Dieudonne, J.; Surles espaces de Kothe, J. Analyse Math. 1 (1951), 81-115. [9] Edwards, R.E.; Functional analysis- Theory and applications, Holt, Rinehart and Winston, New York (1965). [10] Grothendieck, A. ; Sur les espaces (F) et (DF) , Summa Brasil. Math, 3 (1954), 57-123. 246 [11] Horvath, J. ; Topological vector spaces I, Addison­ Wesley Publ. Co.; Reading, Mass. (1966). [12] Husain, T.; The open mapping and closed graph theorems in f;pological vector spaces, Oxford Math. Monographs (1965). [13] Husain, T.; Two new classes of locally convex spaces, Math. Ann. 166 (1966), 289-299. [14] Husain, T .. and Wong, Y.C.; On various types of barrelledness and the hereditary property of (DF)-spaces. Glasgow Math. Journal 17 (1976), 134-143. [15] Husain, T. and Khaleelulla, S.M.; Countably order­ quasibarrelled vector lattices and spaces, Mathematicae Japo.nicae 20 (1975), 3-15.
    [Show full text]