An Introduction to Locally Convex Cones
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On the Cone of a Finite Dimensional Compact Convex Set at a Point
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector On the Cone of a Finite Dimensional Compact Convex Set at a Point C. H. Sung Department of Mathematics University of Califmia at San Diego La Joll41, California 92093 and Department of Mathematics San Diego State University San Diego, California 92182 and B. S. Tam* Department of Mathematics Tamkang University Taipei, Taiwan 25137, Republic of China Submitted by Robert C. Thompson ABSTRACT Four equivalent conditions for a convex cone in a Euclidean space to be an F,-set are given. Our result answers in the negative a recent open problem posed by Tam [5], characterizes the barrier cone of a convex set, and also provides an alternative proof for the known characterizations of the inner aperture of a convex set as given by BrBnsted [2] and Larman [3]. 1. INTRODUCTION Let V be a Euclidean space and C be a nonempty convex set in V. For each y E C, the cone of C at y, denoted by cone(y, C), is the cone *The work of this author was supported in part by the National Science Council of the Republic of China. LINEAR ALGEBRA AND ITS APPLICATIONS 90:47-55 (1987) 47 0 Elsevier Science Publishing Co., Inc., 1987 52 Vanderbilt Ave., New York, NY 10017 00243795/87/$3.50 48 C. H. SUNG AND B. S. TAM { a(r - Y) : o > 0 and x E C} (informally called a point’s cone). On 23 August 1983, at the International Congress of Mathematicians in Warsaw, in a short communication session, the second author presented the paper [5] and posed the following open question: Is it true that, given any cone K in V, there always exists a compact convex set C whose point’s cone at some given point y is equal to K? In view of the relation cone(y, C) = cone(0, C - { y }), the given point y may be taken to be the origin of the space. -
On the Topology of Compactoid Convergence in Non-Archimedean Spaces Annales Mathématiques Blaise Pascal, Tome 3, No 2 (1996), P
ANNALES MATHÉMATIQUES BLAISE PASCAL A.K. KATSARAS A. BELOYIANNIS On the topology of compactoid convergence in non-archimedean spaces Annales mathématiques Blaise Pascal, tome 3, no 2 (1996), p. 135-153 <http://www.numdam.org/item?id=AMBP_1996__3_2_135_0> © Annales mathématiques Blaise Pascal, 1996, tous droits réservés. L’accès aux archives de la revue « Annales mathématiques Blaise Pascal » (http: //math.univ-bpclermont.fr/ambp/) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Math. Blaise Pascal, Vol. 3, N° 2, 1996, pp.135-153 ON THE TOPOLOGY OF COMPACTOID CONVERGENCE IN NON-ARCHIMEDEAN SPACES A. K. KATSARAS and A. BELOYIANNIS Department of Mathematics, University of Ioannina P.O. Box 1186, 451 10 loannina, Greece email: [email protected] Abstract Some of the properties, of the topology of uniform convergence on the compactoid subsets of a non-Archimedean locally convex space E, are studied. In case E is metrizable, the compactoid convergence topology coincides with the finest locally convex topology which agrees with a~E’, E) on equicontinuous sets. 1 1 Introduction In [7J some of the properties of the topology of uniform convergence on the compactoid subsets, of a non-Archimedean locally convex space, are inves- tigated. In the same paper, the authors defined the ~-product E~F of two non-Archimedean locally convex spaces E and F. -
Sisältö Part I: Topological Vector Spaces 4 1. General Topological
Sisalt¨ o¨ Part I: Topological vector spaces 4 1. General topological vector spaces 4 1.1. Vector space topologies 4 1.2. Neighbourhoods and filters 4 1.3. Finite dimensional topological vector spaces 9 2. Locally convex spaces 10 2.1. Seminorms and semiballs 10 2.2. Continuous linear mappings in locally convex spaces 13 3. Generalizations of theorems known from normed spaces 15 3.1. Separation theorems 15 Hahn and Banach theorem 17 Important consequences 18 Hahn-Banach separation theorem 19 4. Metrizability and completeness (Fr`echet spaces) 19 4.1. Baire's category theorem 19 4.2. Complete topological vector spaces 21 4.3. Metrizable locally convex spaces 22 4.4. Fr´echet spaces 23 4.5. Corollaries of Baire 24 5. Constructions of spaces from each other 25 5.1. Introduction 25 5.2. Subspaces 25 5.3. Factor spaces 26 5.4. Product spaces 27 5.5. Direct sums 27 5.6. The completion 27 5.7. Projektive limits 28 5.8. Inductive locally convex limits 28 5.9. Direct inductive limits 29 6. Bounded sets 29 6.1. Bounded sets 29 6.2. Bounded linear mappings 31 7. Duals, dual pairs and dual topologies 31 7.1. Duals 31 7.2. Dual pairs 32 7.3. Weak topologies in dualities 33 7.4. Polars 36 7.5. Compatible topologies 39 7.6. Polar topologies 40 PART II Distributions 42 1. The idea of distributions 42 1.1. Schwartz test functions and distributions 42 2. Schwartz's test function space 43 1 2.1. The spaces C (Ω) and DK 44 1 2 2.2. -
Models of Linear Logic Based on the Schwartz $\Varepsilon $-Product
MODELS OF LINEAR LOGIC BASED ON THE SCHWARTZ ε-PRODUCT. YOANN DABROWSKI AND MARIE KERJEAN Abstract. From the interpretation of Linear Logic multiplicative disjunction as the ε-product de- fined by Laurent Schwartz, we construct several models of Differential Linear Logic based on usual mathematical notions of smooth maps. This improves on previous results in [BET] based on con- venient smoothness where only intuitionist models were built. We isolate a completeness condition, called k-quasi-completeness, and an associated notion stable by duality called k-reflexivity, allow- ing for a ∗-autonomous category of k-reflexive spaces in which the dual of the tensor product is the reflexive version of the ε product. We adapt Meise’s definition of Smooth maps into a first model of Differential Linear Logic, made of k-reflexive spaces. We also build two new models of Linear Logic with conveniently smooth maps, on categories made respectively of Mackey-complete Schwartz spaces and Mackey-complete Nuclear Spaces (with extra reflexivity conditions). Varying slightly the notion of smoothness, one also recovers models of DiLL on the same ∗-autonomous cat- egories. Throughout the article, we work within the setting of Dialogue categories where the tensor product is exactly the ε-product (without reflexivization). Contents 1. Introduction 2 1.1. A first look at the interpretation of Linear Logic constructions 6 Part 1. Three Models of MALL 7 2. Preliminaries 7 2.1. Reminder on topological vector spaces 7 2.2. Reminder on tensor products and duals of locally convex spaces. 8 2.3. Dialogue and ∗-autonomous categories 10 2.4. -
Contents 1. Introduction 1 2. Cones in Vector Spaces 2 2.1. Ordered Vector Spaces 2 2.2
ORDERED VECTOR SPACES AND ELEMENTS OF CHOQUET THEORY (A COMPENDIUM) S. COBZAS¸ Contents 1. Introduction 1 2. Cones in vector spaces 2 2.1. Ordered vector spaces 2 2.2. Ordered topological vector spaces (TVS) 7 2.3. Normal cones in TVS and in LCS 7 2.4. Normal cones in normed spaces 9 2.5. Dual pairs 9 2.6. Bases for cones 10 3. Linear operators on ordered vector spaces 11 3.1. Classes of linear operators 11 3.2. Extensions of positive operators 13 3.3. The case of linear functionals 14 3.4. Order units and the continuity of linear functionals 15 3.5. Locally order bounded TVS 15 4. Extremal structure of convex sets and elements of Choquet theory 16 4.1. Faces and extremal vectors 16 4.2. Extreme points, extreme rays and Krein-Milman's Theorem 16 4.3. Regular Borel measures and Riesz' Representation Theorem 17 4.4. Radon measures 19 4.5. Elements of Choquet theory 19 4.6. Maximal measures 21 4.7. Simplexes and uniqueness of representing measures 23 References 24 1. Introduction The aim of these notes is to present a compilation of some basic results on ordered vector spaces and positive operators and functionals acting on them. A short presentation of Choquet theory is also included. They grew up from a talk I delivered at the Seminar on Analysis and Optimization. The presentation follows mainly the books [3], [9], [19], [22], [25], and [11], [23] for the Choquet theory. Note that the first two chapters of [9] contains a thorough introduction (with full proofs) to some basics results on ordered vector spaces. -
A Short Description on Locally Convex and Polar Topologies
International Journal of Engineering Science Invention (IJESI) ISSN (Online): 2319-6734, ISSN (Print): 2319-6726 www.ijesi.org ||Volume 9 Issue 7 Series II || July 2020 || PP 01-05 A Short Description on Locally Convex and Polar Topologies Dr. Krishnandan Prasad1, Dr. Amar Nath Kumar2 1(Mathematics,T.P.S. College/ Patliputra University,India) 2(Mathematics, Patna Science College/Patna University Name,India) ABSTRACT: In this paper an attempt has been made to make some positive contributions to the study of topological dynamical systems. It is well known that the classical studies of dynamical systems are based on Hamilton’s principle and the principle of least action and techniques employed in them is usually the variational calculus. Some years ago, studies has been made of dynamical systems which carry topologies on them making the dynamical behavior of the system continuous i.e. the so-called topologies campactible with dynamical structure of the system. This relates the study of dynamical systems with the study of continuous semigroup of operators on Banach Algebras, specially C*-Algebra, and W*-Algebras. The most useful topologies for such studies are so-called weak topologies, strong topologies and the corresponding notations of weak continuities and strong continuities. Thus we are lead to the studies of topological dynamical systems in the context of topological vector space, specially locally convex spaces and locally compact groups. KEYWORDS - Absorbent, Absolutely Convex, Balanced, Convex, , Locally Convex Topological Linear -
Marie Kerjean
Tensor products and *-autonomous categories Marie Kerjean Laboratory PPS, Universit´eParis Diderot [email protected] The main use of ∗-autonomous categories is in the semantic study of Linear Logic. For this reason, it is thus natural to look for a ∗-autonomous category of locally convex topological vector spaces (tvs). On one hand, Linear Logic inherits its semantics from Linear Algebra, and it is thus natural to build models of Linear Logic from vector spaces [3,5,6,4]. On the other hand, denotational semantics has sought continuous models of computation through Scott domains [9]. Moreover, the infinite nature of the exponential of Linear Logic calls for infinite dimensional spaces, for which topology becomes necessary. One of the first intuitions that comes to mind when thinking about objects in a ∗-autonomous category is the notion of reflexive vector space, i.e. a a tvs which equals its double dual. When A is a vector space, the transpose dA : A ! (A !?) !? of the evaluation map evA :(A !?) × A !? is exactly the canonical injection of a vector space in its bidual. Then, requiring dA to be an isomorphism amounts to requiring A to be reflexive. However, the category of reflexive topological vector spaces is not ∗-autonomous, as it is not closed. Barr [2] constructs two closed subcategories of the category of tvs by re- stricting to tvs endowed with their weak topology (wtvs) or with their Mackey topology (mtvs), which are both polar topologies. Indeed, if E is a tvs, one can define its dual E0 as the space of all continuous linear form on E. -
Some Applications of the Hahn-Banach Separation Theorem 1
Some Applications of the Hahn-Banach Separation Theorem 1 Frank Deutsch , Hein Hundal, Ludmil Zikatanov January 1, 2018 arXiv:1712.10250v1 [math.FA] 29 Dec 2017 1We dedicate this paper to our friend and colleague Professor Yair Censor on the occasion of his 75th birthday. Abstract We show that a single special separation theorem (namely, a consequence of the geometric form of the Hahn-Banach theorem) can be used to prove Farkas type theorems, existence theorems for numerical quadrature with pos- itive coefficients, and detailed characterizations of best approximations from certain important cones in Hilbert space. 2010 Mathematics Subject Classification: 41A65, 52A27. Key Words and Phrases: Farkas-type theorems, numerical quadrature, characterization of best approximations from convex cones in Hilbert space. 1 Introduction We show that a single separation theorem|the geometric form of the Hahn- Banach theorem|has a variety of different applications. In section 2 we state this general separation theorem (Theorem 2.1), but note that only a special consequence of it is needed for our applications (Theorem 2.2). The main idea in Section 2 is the notion of a functional being positive relative to a set of functionals (Definition 2.3). Then a useful characterization of this notion is given in Theorem 2.4. Some applications of this idea are given in Section 3. They include a proof of the existence of numerical quadrature with positive coefficients, new proofs of Farkas type theorems, an application to determining best approximations from certain convex cones in Hilbert space, and a specific application of the latter to determine best approximations that are also shape-preserving. -
Complemented Brunn–Minkowski Inequalities and Isoperimetry for Homogeneous and Non-Homogeneous Measures
Complemented Brunn{Minkowski Inequalities and Isoperimetry for Homogeneous and Non-Homogeneous Measures Emanuel Milman1 and Liran Rotem2 Abstract Elementary proofs of sharp isoperimetric inequalities on a normed space (Rn; k·k) equipped with a measure µ = w(x)dx so that wp is homogeneous are provided, along with a characterization of the corresponding equality cases. When p 2 (0; 1] and in addition wp is assumed concave, the result is an immediate corollary of the Borell{Brascamp{Lieb extension of the classical Brunn{Minkowski inequality, providing a new elementary proof of a recent result of Cabr´e{RosOton{Serra. When p 2 (−1=n; 0), the relevant property turns out to be a novel \q{complemented Brunn{Minkowski" inequality: n n n 8λ 2 (0; 1) 8 Borel sets A; B ⊂ R such that µ(R n A); µ(R n B) < 1 ; n n q n q 1=q µ∗(R n (λA + (1 − λ)B)) ≤ (λµ(R n A) + (1 − λ)µ(R n B) ) ; p 1 1 which we show is always satisfied by µ when w is homogeneous with q = p + n; in par- ticular, this is satisfied by the Lebesgue measure with q = 1=n. This gives rise to a new class of measures, which are \complemented" analogues of the class of convex measures introduced by Borell, but which have vastly different properties. The resulting isoperi- metric inequality and characterization of isoperimetric minimizers extends beyond the re- cent results of Ca~nete{Rosalesand Howe. The isoperimetric and Brunn-Minkowski type inequalities also extend to the non-homogeneous setting, under a certain log-convexity assumption on the density. -
Polar Locally Convex Topologies and Attouch-Wets Convergence
BULL. AUSTRAL. MATH. SOC. 46AO5, 46A20, 54B20 VOL. 47 (1993) [33-46] POLAR LOCALLY CONVEX TOPOLOGIES AND ATTOUCH-WETS CONVERGENCE GERALD BEER Let X be a Hausdorff locally convex space. We show that convergence of a net of continuous linear functionals on X with respect to a given polar topology on its continuous dual X' can be explained in terms of the convergence of the cor- responding net of its graphs in X X R, and the corresponding net of level sets at a fixed height in X, with respect to a natural generalisation of Attouch-Wets convergence in normable spaces. 1. INTRODUCTION Let X be a real normed linear space and let 3/, yl, 2/2 > 2/3 j • • • be continuous linear functionals on X. It has been long known that convergence of (yn) to y in the norm topology can be explained in terms of the convergence of the associated sequence of graphs (Grt/n) to Gry in X x R, with respect to a metric 6 on the closed linear subspaces of a normed space given by Kato [10, p.197-204]. For closed subspaces M and N, this distance may be described by 6{M, N) = inf{e >0: M nU cN+eU and NnU C M + eU}, where U is the solid unit ball of the space. Evidently, this notion of distance is not appropriate for the more general class of closed convex sets. For one thing, such sets need not hit the unit ball. Furthermore, even for convex sets that do hit the unit ball, the triangle inequality may fail with respect to this notion of distance. -
The Banach-Steinhaus Theorem in Abstract Duality Pairs
Proyecciones Journal of Mathematics Vol. 34, No 4, pp. 391-399, December 2015. Universidad Cat´olica del Norte Antofagasta - Chile The Banach-Steinhaus Theorem in Abstract Duality Pairs Min-Hyung Cho Kum-Oh National Institute of Technology, Korea Li Ronglu Harbin Institute of Technology, P. R. C. and Charles Swartz New Mexico State University, U. S. A. Received : October 2015. Accepted : November 2015 Abstract Let E,F be sets and G a Hausdorff, abelian topological group with b : E F G;werefertoE,F,G as an abstract duality pair with respect× to G→or an abstract triple and denote this by (E,F : G).Let (Ei,Fi : G) be abstract triples for i =1, 2.Let i be a family of F subsets of Fi and let τ i (Ei)=τi be the topology on Ei of uniform F convergence on the members of i.LetΓ be a family of mappings F from E1 to E2. We consider conditions which guarantee that Γ is τ1 τ2 equicontinuous. We then apply the results to obtain versions of− the Banach-Steinhaus Theorem for both abstract triples and for linear operators between locally convex spaces. 392 Min-Hyung Cho, Li Ronglu and Charles Swartz In [CLS] we established versions of the Orlicz-Pettis Theorem for sub- series convergent series in abstract triples or abstract duality pairs based on results which were initiated at New Mexico State University during Pro- fessor Li Ronglu’s tenure as a visiting scholar. In this note we present some further results on an equicontinuity version of the Banach-Steinhaus The- orem for abstract triples which were also the result of Professor Li’s visit. -
Conic Linear Programming
Conic Linear Programming Yinyu Ye December 2004, revised January 2015 i ii Preface This monograph is developed for MS&E 314, \Conic Linear Programming", which I am teaching at Stanford. Information, lecture slides, supporting mate- rials, and computer programs related to this book may be found at the following address on the World-Wide Web: http://www.stanford.edu/class/msande314 Please report any question, comment and error to the address: [email protected] A little story in the development of semidefinite programming (SDP), a major subclass of conic linear programming. One day in 1990, I visited the Computer Science Department of the University of Minnesota and met a young graduate student, Farid Alizadeh. He, working then on combinatorial optimiza- tion, introduced me “semidefinite optimization" or linear programming over the positive definite matrix cone. We had a very extensive discussion that afternoon and concluded that interior-point linear programming algorithms could be ap- plicable to solving SDPs. I suggested Farid to look at the linear programming (LP) interior-point algorithms and to develop an SDP (primal) potential reduc- tion algorithm. He worked hard for several months, and one afternoon showed up in my office in Iowa City, about 300 miles from Minneapolis. He had every- thing worked out, including potential function, algorithm, complexity bound, and even a \dictionary" list between LP and SDP. But he was stuck on one problem that was on how to keep the symmetry of the scaled directional ma- trix. We went to a bar nearby on Clinton Street in Iowa City (I paid for him since I was a third-year professor then and eager to demonstrate that I could take care of my students).