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Conjugate Convex Functions in Topological Vector Spaces
Matematisk-fysiske Meddelelser udgivet af Det Kongelige Danske Videnskabernes Selskab Bind 34, nr. 2 Mat. Fys. Medd . Dan. Vid. Selsk. 34, no. 2 (1964) CONJUGATE CONVEX FUNCTIONS IN TOPOLOGICAL VECTOR SPACES BY ARNE BRØNDSTE D København 1964 Kommissionær : Ejnar Munksgaard Synopsis Continuing investigations by W. L . JONES (Thesis, Columbia University , 1960), the theory of conjugate convex functions in finite-dimensional Euclidea n spaces, as developed by W. FENCHEL (Canadian J . Math . 1 (1949) and Lecture No- tes, Princeton University, 1953), is generalized to functions in locally convex to- pological vector spaces . PRINTP_ll IN DENMARK BIANCO LUNOS BOGTRYKKERI A-S Introduction The purpose of the present paper is to generalize the theory of conjugat e convex functions in finite-dimensional Euclidean spaces, as initiated b y Z . BIRNBAUM and W. ORLICz [1] and S . MANDELBROJT [8] and developed by W. FENCHEL [3], [4] (cf. also S. KARLIN [6]), to infinite-dimensional spaces . To a certain extent this has been done previously by W . L . JONES in his Thesis [5] . His principal results concerning the conjugates of real function s in topological vector spaces have been included here with some improve- ments and simplified proofs (Section 3). After the present paper had bee n written, the author ' s attention was called to papers by J . J . MOREAU [9], [10] , [11] in which, by a different approach and independently of JONES, result s equivalent to many of those contained in this paper (Sections 3 and 4) are obtained. Section 1 contains a summary, based on [7], of notions and results fro m the theory of topological vector spaces applied in the following . -
Arxiv:1702.07867V1 [Math.FA]
TOPOLOGICAL PROPERTIES OF STRICT (LF )-SPACES AND STRONG DUALS OF MONTEL STRICT (LF )-SPACES SAAK GABRIYELYAN Abstract. Following [2], a Tychonoff space X is Ascoli if every compact subset of Ck(X) is equicontinuous. By the classical Ascoli theorem every k- space is Ascoli. We show that a strict (LF )-space E is Ascoli iff E is a Fr´echet ′ space or E = ϕ. We prove that the strong dual Eβ of a Montel strict (LF )- space E is an Ascoli space iff one of the following assertions holds: (i) E is a ′ Fr´echet–Montel space, so Eβ is a sequential non-Fr´echet–Urysohn space, or (ii) ′ ω D E = ϕ, so Eβ = R . Consequently, the space (Ω) of test functions and the space of distributions D′(Ω) are not Ascoli that strengthens results of Shirai [20] and Dudley [5], respectively. 1. Introduction. The class of strict (LF )-spaces was intensively studied in the classic paper of Dieudonn´eand Schwartz [3]. It turns out that many of strict (LF )-spaces, in particular a lot of linear spaces considered in Schwartz’s theory of distributions [18], are not metrizable. Even the simplest ℵ0-dimensional strict (LF )-space ϕ, n the inductive limit of the sequence {R }n∈ω, is not metrizable. Nyikos [16] showed that ϕ is a sequential non-Fr´echet–Urysohn space (all relevant definitions are given in the next section). On the other hand, Shirai [20] proved the space D(Ω) of test functions over an open subset Ω of Rn, which is one of the most famous example of strict (LF )-spaces, is not sequential. -
S-Barrelled Topological Vector Spaces*
Canad. Math. Bull. Vol. 21 (2), 1978 S-BARRELLED TOPOLOGICAL VECTOR SPACES* BY RAY F. SNIPES N. Bourbaki [1] was the first to introduce the class of locally convex topological vector spaces called "espaces tonnelés" or "barrelled spaces." These spaces have some of the important properties of Banach spaces and Fréchet spaces. Indeed, a generalized Banach-Steinhaus theorem is valid for them, although barrelled spaces are not necessarily metrizable. Extensive accounts of the properties of barrelled locally convex topological vector spaces are found in [5] and [8]. In the last few years, many new classes of locally convex spaces having various types of barrelledness properties have been defined (see [6] and [7]). In this paper, using simple notions of sequential convergence, we introduce several new classes of topological vector spaces which are similar to Bourbaki's barrelled spaces. The most important of these we call S-barrelled topological vector spaces. A topological vector space (X, 2P) is S-barrelled if every sequen tially closed, convex, balanced, and absorbing subset of X is a sequential neighborhood of the zero vector 0 in X Examples are given of spaces which are S-barrelled but not barrelled. Then some of the properties of S-barrelled spaces are given—including permanence properties, and a form of the Banach- Steinhaus theorem valid for them. S-barrelled spaces are useful in the study of sequentially continuous linear transformations and sequentially continuous bilinear functions. For the most part, we use the notations and definitions of [5]. Proofs which closely follow the standard patterns are omitted. 1. Definitions and Examples. -
Recent Developments in the Theory of Duality in Locally Convex Vector Spaces
[ VOLUME 6 I ISSUE 2 I APRIL– JUNE 2019] E ISSN 2348 –1269, PRINT ISSN 2349-5138 RECENT DEVELOPMENTS IN THE THEORY OF DUALITY IN LOCALLY CONVEX VECTOR SPACES CHETNA KUMARI1 & RABISH KUMAR2* 1Research Scholar, University Department of Mathematics, B. R. A. Bihar University, Muzaffarpur 2*Research Scholar, University Department of Mathematics T. M. B. University, Bhagalpur Received: February 19, 2019 Accepted: April 01, 2019 ABSTRACT: : The present paper concerned with vector spaces over the real field: the passage to complex spaces offers no difficulty. We shall assume that the definition and properties of convex sets are known. A locally convex space is a topological vector space in which there is a fundamental system of neighborhoods of 0 which are convex; these neighborhoods can always be supposed to be symmetric and absorbing. Key Words: LOCALLY CONVEX SPACES We shall be exclusively concerned with vector spaces over the real field: the passage to complex spaces offers no difficulty. We shall assume that the definition and properties of convex sets are known. A convex set A in a vector space E is symmetric if —A=A; then 0ЄA if A is not empty. A convex set A is absorbing if for every X≠0 in E), there exists a number α≠0 such that λxЄA for |λ| ≤ α ; this implies that A generates E. A locally convex space is a topological vector space in which there is a fundamental system of neighborhoods of 0 which are convex; these neighborhoods can always be supposed to be symmetric and absorbing. Conversely, if any filter base is given on a vector space E, and consists of convex, symmetric, and absorbing sets, then it defines one and only one topology on E for which x+y and λx are continuous functions of both their arguments. -
On Nonbornological Barrelled Spaces Annales De L’Institut Fourier, Tome 22, No 2 (1972), P
ANNALES DE L’INSTITUT FOURIER MANUEL VALDIVIA On nonbornological barrelled spaces Annales de l’institut Fourier, tome 22, no 2 (1972), p. 27-30 <http://www.numdam.org/item?id=AIF_1972__22_2_27_0> © Annales de l’institut Fourier, 1972, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Inst. Fourier, Grenoble 22, 2 (1972), 27-30. ON NONBORNOLOGICAL BARRELLED SPACES 0 by Manuel VALDIVIA L. Nachbin [5] and T. Shirota [6], give an answer to a problem proposed by N. Bourbaki [1] and J. Dieudonne [2], giving an example of a barrelled space, which is not bornological. Posteriorly some examples of nonbornological barrelled spaces have been given, e.g. Y. Komura, [4], has constructed a Montel space which is not bornological. In this paper we prove that if E is the topological product of an uncountable family of barrelled spaces, of nonzero dimension, there exists an infinite number of barrelled subspaces of E, which are not bornological.We obtain also an analogous result replacing « barrelled » by « quasi-barrelled ». We use here nonzero vector spaces on the field K of real or complex number. The topologies on these spaces are sepa- rated. -
On L−Generalized Filters and Nets
Annals of Fuzzy Mathematics and Informatics Volume x, No. x, (Month 201y), pp. 1{xx @FMI ISSN: 2093{9310 (print version) c Kyung Moon Sa Co. ISSN: 2287{6235 (electronic version) http://www.kyungmoon.com http://www.afmi.or.kr On L−generalized filters and nets G. A. Kamel Received 13 March 2017; Revised 20 April 2017; Accepted 5 May 2017 Abstract. In this paper, we consider the L−generalized filters and L−generalized nets on the non empty universal set X to be a natural generalization of the crisp generalized filters and the crisp generalized nets respectively. In each the crisp and L−generalized cases, the extended and restricted filters (nets) will be obtained respectively. The construction of convergence to limit points of the crisp generalized and L−generalized filters (nets) will be studied respectively. We also show that any crisp generalized net on the set X associates a unique L−generalized net on X; and in general the converse is not true. In addition to that we also show that for any given L−generalized filter there exists the L−generalized net, and the vice-versa is true. Finally, the relations between the convergence of the L−generalized filters and the L−generalized nets will be outlined. Moreover, to support these concepts and relations, the construction of some examples in all sections of paper will be studied. 2010 AMS Classification: 54A40 Keywords: Generalized filter, Generalized nets, L−generalized filter, L−generalized nets, Neighbourhoods generalized filter systems. Corresponding Author: G. A. Kamel ([email protected]) 1. Introduction Zadeh in his famous paper [23] introduced the concept of a fuzzy set A of the non-empty universal set X as a mapping A : X −! [0; 1]: Chang in [4] had first tried to fuzzify the concept of a topology on a non-empty set X as a subfamily of [0; 1]X , satisfying the same axioms of ordinary topology. -
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]. -
Chapter 1: Bases for Topologies Contents
MA3421 2016–17 Chapter 1: Bases for topologies Contents 1.1 Review: metric and topological spaces . .2 1.2 Review: continuity . .7 1.3 Review: limits of sequences . .8 1.4 Base for a topology . 11 1.5 Sub-bases, weak and product topologies . 14 1.6 Neighbourhood bases . 21 1 2 Chapter 1: Bases for topologies 1.1 Review: metric and topological spaces From MA2223 last year, you should know what a metric space is and what the metric topology is. Here is a quick refresher. Definition 1.1.1. Given any set X of points and a function d: X × X ! [0; 1) ⊂ R with these 3 properties: (i) d(z; w) ≥ 0 with equality if and only if z = w; (ii) d(z; w) = d(w; z); (iii) d(z; w) ≤ d(z; v) + d(v; w) (triangle inequality), we say that d is a metric on the space X and we call the combination (X; d) a metric space. Examples 1.1.2. Rn will denote the usual n-dimensional space (over R) and Cn the complex version. We define the (standard) Euclidean distance between pairs of points in Rn by v u n uX 2 d(x; y) = t (xj − yj) j=1 2 3 (abstracting the distance formula from R or R ), for x = (x1; x2; : : : ; xn) and y = (y1; y2; : : : ; yn). For n = 1, recall d(x; y) = jx − yj. n In the case on n-tuples z = (z1; z2; : : : ; zn) and w = (w1; w2; : : : ; wn) 2 C , we also define the standard distance via v u n uX 2 d(z; w) = t jzj − wjj j=1 Recall that for z = x + iy 2 C (with x; y 2 R the real and imaginary parts of z and i2 = −1) the modulus (or absolute value) of such a z is jzj = px2 + y2. -
Arxiv:1712.05521V4 [Math.GN]
MAXIMALLY ALMOST PERIODIC GROUPS AND RESPECTING PROPERTIES SAAK GABRIYELYAN Abstract. For a Tychonoff space X, denote by P the family of topological properties P of being a convergent sequence or being a compact, sequentially compact, countably compact, pseudocompact and functionally bounded subset of X, respectively. A maximally almost periodic (MAP ) group G respects P if P(G)= P(G+), where G+ is the group G endowed with the Bohr topology. We study relations between different respecting properties from P and show that the respecting convergent sequences (=the Schur property) is the weakest one among the properties of P. We characterize respecting properties from P in wide classes of MAP topological groups including the class of metrizable MAP abelian groups. Every real locally convex space (lcs) is a quotient space of an lcs with the Schur property, and every locally quasi-convex (lqc) abelian group is a quotient group of an lqc abelian group with the Schur property. It is shown that a reflexive group G has the Schur ∧ property or respects compactness iff its dual group G is c0-barrelled or g-barrelled, respectively. We prove that an lqc abelian kω-group respects all properties P ∈ P. As an application of the obtained results we show that (1) the space Ck(X) is a reflexive group for every separable metrizable space X, and (2) a reflexive abelian group of finite exponent is a Mackey group. 1. Introduction Let X be a Tychonoff space. If P is a topological property, we denote by P(X) the set of all subspaces of X with P. -
Locally Convex Spaces Manv 250067-1, 5 Ects, Summer Term 2017 Sr 10, Fr
LOCALLY CONVEX SPACES MANV 250067-1, 5 ECTS, SUMMER TERM 2017 SR 10, FR. 13:15{15:30 EDUARD A. NIGSCH These lecture notes were developed for the topics course locally convex spaces held at the University of Vienna in summer term 2017. Prerequisites consist of general topology and linear algebra. Some background in functional analysis will be helpful but not strictly mandatory. This course aims at an early and thorough development of duality theory for locally convex spaces, which allows for the systematic treatment of the most important classes of locally convex spaces. Further topics will be treated according to available time as well as the interests of the students. Thanks for corrections of some typos go out to Benedict Schinnerl. 1 [git] • 14c91a2 (2017-10-30) LOCALLY CONVEX SPACES 2 Contents 1. Introduction3 2. Topological vector spaces4 3. Locally convex spaces7 4. Completeness 11 5. Bounded sets, normability, metrizability 16 6. Products, subspaces, direct sums and quotients 18 7. Projective and inductive limits 24 8. Finite-dimensional and locally compact TVS 28 9. The theorem of Hahn-Banach 29 10. Dual Pairings 34 11. Polarity 36 12. S-topologies 38 13. The Mackey Topology 41 14. Barrelled spaces 45 15. Bornological Spaces 47 16. Reflexivity 48 17. Montel spaces 50 18. The transpose of a linear map 52 19. Topological tensor products 53 References 66 [git] • 14c91a2 (2017-10-30) LOCALLY CONVEX SPACES 3 1. Introduction These lecture notes are roughly based on the following texts that contain the standard material on locally convex spaces as well as more advanced topics. -
Functional Properties of Hörmander's Space of Distributions Having A
Functional properties of Hörmander’s space of distributions having a specified wavefront set Yoann Dabrowski, Christian Brouder To cite this version: Yoann Dabrowski, Christian Brouder. Functional properties of Hörmander’s space of distributions having a specified wavefront set. 2014. hal-00850192v2 HAL Id: hal-00850192 https://hal.archives-ouvertes.fr/hal-00850192v2 Preprint submitted on 3 May 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Communications in Mathematical Physics manuscript No. (will be inserted by the editor) Functional properties of H¨ormander’s space of distributions having a specified wavefront set Yoann Dabrowski1, Christian Brouder2 1 Institut Camille Jordan UMR 5208, Universit´ede Lyon, Universit´eLyon 1, 43 bd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France 2 Institut de Min´eralogie, de Physique des Mat´eriaux et de Cosmochimie, Sorbonne Univer- sit´es, UMR CNRS 7590, UPMC Univ. Paris 06, Mus´eum National d’Histoire Naturelle, IRD UMR 206, 4 place Jussieu, F-75005 Paris, France. Received: date / Accepted: date ′ Abstract: The space Γ of distributions having their wavefront sets in a closed cone Γ has become importantD in physics because of its role in the formulation of quantum field theory in curved spacetime. -
Topology Proceedings
Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: [email protected] ISSN: 0146-4124 COPYRIGHT °c by Topology Proceedings. All rights reserved. TOPOLOGY PROCEEDINGS Volume 27, No. 2, 2003 Pages 661{673 CONNECTED URYSOHN SUBTOPOLOGIES RICHARD G. WILSON∗ Abstract. We show that each second countable Urysohn space which is not Urysohn-closed can be condensed onto a connected Urysohn space and as a corollary we charac- terize those countable Urysohn spaces which have connected Urysohn subtopologies. We also answer two questions from [12] regarding condensations onto connected Hausdorff spaces. 1. Introduction and Preliminary Results Recall that a space is Urysohn if distinct points have disjoint closed neighbourhoods and a space is feebly compact if every locally finite family of open sets is finite. During the 1960's and 70's many papers appeared in which countable connected Urysohn spaces were constructed (see for example [6],[7],[8] and [9]); such spaces were in some sense considered oddities. In a previous paper [12], we proved that each disconnected Hausdorff space with a countable network can be condensed (that is to say, there is a continuous bijection) onto a connected Hausdorff space if and only if it is not feebly compact. Here we prove an analogous result for Urysohn spaces: 2000 Mathematics Subject Classification. Primary 54D05; Secondary 54D10, 54D25. Key words and phrases. Second countable connected Urysohn space, count- able network, condensation, Urysohn family, Urysohn filter, Urysohn-closed space. ∗Research supported by Consejo Nacional de Ciencia y Tecnolog´ıa(CONA- CYT), Mexico, grant no.