Notes, We Review Useful Concepts in Convexity and Normed Spaces
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Arxiv:Math/9201256V1 [Math.RT] 1 Jan 1992 Naso Lblaayi N Geometry and Vol
Annals of Global Analysis and Geometry Vol. 8, No. 3 (1990), 299–313 THE MOMENT MAPPING FOR UNITARY REPRESENTATIONS Peter W. Michor Institut f¨ur Mathematik der Universit¨at Wien, Austria Abstract. For any unitary representation of an arbitrary Lie group I construct a moment mapping from the space of smooth vectors of the representation into the dual of the Lie algebra. This moment mapping is equivariant and smooth. For the space of analytic vectors the same construction is possible and leads to a real analytic moment mapping. Table of contents 1.Introduction . .. .. .. .. .. .. .. .. .. .. 1 2.Calculusofsmoothmappings . 2 3.Calculusofholomorphicmappings . 5 4.Calculusofrealanalyticmappings . 6 5.TheSpaceofSmoothVectors . 7 6.Themodelforthemomentmapping . 8 7. Hamiltonian Mechanics on H∞ .................... 9 8. The moment mapping for a unitary representation . .... 11 9.Therealanalyticmomentmapping . 13 1. Introduction arXiv:math/9201256v1 [math.RT] 1 Jan 1992 With the help of the cartesian closed calculus for smooth mappings as explained in [F-K] we can show, that for any Lie group and for any unitary representation its restriction to the space of smooth vectors is smooth. The imaginary part of the hermitian inner product restricts to a ”weak” symplectic structure on the vector space of smooth vectors. This gives rise to the Poisson bracket on a suitably chosen space of smooth functions on the space of smooth vectors. The derivative of the representation on the space of smooth vectors is a symplectic action of the Lie algebra, which can be lifted to a Hamiltonian action, i.e. a Lie algebra homomorphism from the Lie algebra into the function space with the Poisson bracket. -
The Hahn-Banach Theorem and Infima of Convex Functions
The Hahn-Banach theorem and infima of convex functions Hajime Ishihara School of Information Science Japan Advanced Institute of Science and Technology (JAIST) Nomi, Ishikawa 923-1292, Japan first CORE meeting, LMU Munich, 7 May, 2016 Normed spaces Definition A normed space is a linear space E equipped with a norm k · k : E ! R such that I kxk = 0 $ x = 0, I kaxk = jajkxk, I kx + yk ≤ kxk + kyk, for each x; y 2 E and a 2 R. Note that a normed space E is a metric space with the metric d(x; y) = kx − yk: Definition A Banach space is a normed space which is complete with respect to the metric. Examples For 1 ≤ p < 1, let p N P1 p l = f(xn) 2 R j n=0 jxnj < 1g and define a norm by P1 p 1=p k(xn)k = ( n=0 jxnj ) : Then lp is a (separable) Banach space. Examples Classically the normed space 1 N l = f(xn) 2 R j (xn) is boundedg with the norm k(xn)k = sup jxnj n is an inseparable Banach space. However, constructively, it is not a normed space. Linear mappings Definition A mapping T between linear spaces E and F is linear if I T (ax) = aTx, I T (x + y) = Tx + Ty for each x; y 2 E and a 2 R. Definition A linear functional f on a linear space E is a linear mapping from E into R. Bounded linear mappings Definition A linear mapping T between normed spaces E and F is bounded if there exists c ≥ 0 such that kTxk ≤ ckxk for each x 2 E. -
Sharp Finiteness Principles for Lipschitz Selections: Long Version
Sharp finiteness principles for Lipschitz selections: long version By Charles Fefferman Department of Mathematics, Princeton University, Fine Hall Washington Road, Princeton, NJ 08544, USA e-mail: [email protected] and Pavel Shvartsman Department of Mathematics, Technion - Israel Institute of Technology, 32000 Haifa, Israel e-mail: [email protected] Abstract Let (M; ρ) be a metric space and let Y be a Banach space. Given a positive integer m, let F be a set-valued mapping from M into the family of all compact convex subsets of Y of dimension at most m. In this paper we prove a finiteness principle for the existence of a Lipschitz selection of F with the sharp value of the finiteness number. Contents 1. Introduction. 2 1.1. Main definitions and main results. 2 1.2. Main ideas of our approach. 3 2. Nagata condition and Whitney partitions on metric spaces. 6 arXiv:1708.00811v2 [math.FA] 21 Oct 2017 2.1. Metric trees and Nagata condition. 6 2.2. Whitney Partitions. 8 2.3. Patching Lemma. 12 3. Basic Convex Sets, Labels and Bases. 17 3.1. Main properties of Basic Convex Sets. 17 3.2. Statement of the Finiteness Theorem for bounded Nagata Dimension. 20 Math Subject Classification 46E35 Key Words and Phrases Set-valued mapping, Lipschitz selection, metric tree, Helly’s theorem, Nagata dimension, Whitney partition, Steiner-type point. This research was supported by Grant No 2014055 from the United States-Israel Binational Science Foundation (BSF). The first author was also supported in part by NSF grant DMS-1265524 and AFOSR grant FA9550-12-1-0425. -
On Order Preserving and Order Reversing Mappings Defined on Cones of Convex Functions MANUSCRIPT
ON ORDER PRESERVING AND ORDER REVERSING MAPPINGS DEFINED ON CONES OF CONVEX FUNCTIONS MANUSCRIPT Lixin Cheng∗ Sijie Luo School of Mathematical Sciences Yau Mathematical Sciences Center Xiamen University Tsinghua University Xiamen 361005, China Beijing 100084, China [email protected] [email protected] ABSTRACT In this paper, we first show that for a Banach space X there is a fully order reversing mapping T from conv(X) (the cone of all extended real-valued lower semicontinuous proper convex functions defined on X) onto itself if and only if X is reflexive and linearly isomorphic to its dual X∗. Then we further prove the following generalized “Artstein-Avidan-Milman” representation theorem: For every fully order reversing mapping T : conv(X) → conv(X) there exist a linear isomorphism ∗ ∗ ∗ U : X → X , x0, ϕ0 ∈ X , α> 0 and r0 ∈ R so that ∗ (Tf)(x)= α(Ff)(Ux + x0)+ hϕ0, xi + r0, ∀x ∈ X, where F : conv(X) → conv(X∗) is the Fenchel transform. Hence, these resolve two open questions. We also show several representation theorems of fully order preserving mappings de- fined on certain cones of convex functions. For example, for every fully order preserving mapping S : semn(X) → semn(X) there is a linear isomorphism U : X → X so that (Sf)(x)= f(Ux), ∀f ∈ semn(X), x ∈ X, where semn(X) is the cone of all lower semicontinuous seminorms on X. Keywords Fenchel transform · order preserving mapping · order reversing mapping · convex function · Banach space 1 Introduction An elegant theorem of Artstein-Avidan and Milman [5] states that every fully order reversing (resp. -
The Continuity of Additive and Convex Functions, Which Are Upper Bounded
THE CONTINUITY OF ADDITIVE AND CONVEX FUNCTIONS, WHICH ARE UPPER BOUNDED ON NON-FLAT CONTINUA IN Rn TARAS BANAKH, ELIZA JABLO NSKA,´ WOJCIECH JABLO NSKI´ Abstract. We prove that for a continuum K ⊂ Rn the sum K+n of n copies of K has non-empty interior in Rn if and only if K is not flat in the sense that the affine hull of K coincides with Rn. Moreover, if K is locally connected and each non-empty open subset of K is not flat, then for any (analytic) non-meager subset A ⊂ K the sum A+n of n copies of A is not meager in Rn (and then the sum A+2n of 2n copies of the analytic set A has non-empty interior in Rn and the set (A − A)+n is a neighborhood of zero in Rn). This implies that a mid-convex function f : D → R, defined on an open convex subset D ⊂ Rn is continuous if it is upper bounded on some non-flat continuum in D or on a non-meager analytic subset of a locally connected nowhere flat subset of D. Let X be a linear topological space over the field of real numbers. A function f : X → R is called additive if f(x + y)= f(x)+ f(y) for all x, y ∈ X. R x+y f(x)+f(y) A function f : D → defined on a convex subset D ⊂ X is called mid-convex if f 2 ≤ 2 for all x, y ∈ D. Many classical results concerning additive or mid-convex functions state that the boundedness of such functions on “sufficiently large” sets implies their continuity. -
Convenient Vector Spaces, Convenient Manifolds and Differential Linear Logic
Convenient Vector Spaces, Convenient Manifolds and Differential Linear Logic Richard Blute University of Ottawa ongoing discussions with Geoff Crutwell, Thomas Ehrhard, Alex Hoffnung, Christine Tasson June 20, 2011 Richard Blute Convenient Vector Spaces, Convenient Manifolds and Differential Linear Logic Goals Develop a theory of (smooth) manifolds based on differential linear logic. Or perhaps develop a differential linear logic based on manifolds. Convenient vector spaces were recently shown to be a model. There is a well-developed theory of convenient manifolds, including infinite-dimensional manifolds. Convenient manifolds reveal additional structure not seen in finite dimensions. In particular, the notion of tangent space is much more complex. Synthetic differential geometry should also provide information. Convenient vector spaces embed into an extremely good model. Richard Blute Convenient Vector Spaces, Convenient Manifolds and Differential Linear Logic Convenient vector spaces (Fr¨olicher,Kriegl) Definition A vector space is locally convex if it is equipped with a topology such that each point has a neighborhood basis of convex sets, and addition and scalar multiplication are continuous. Locally convex spaces are the most well-behaved topological vector spaces, and most studied in functional analysis. Note that in any topological vector space, one can take limits and hence talk about derivatives of curves. A curve is smooth if it has derivatives of all orders. The analogue of Cauchy sequences in locally convex spaces are called Mackey-Cauchy sequences. The convergence of Mackey-Cauchy sequences implies the convergence of all Mackey-Cauchy nets. The following is taken from a long list of equivalences. Richard Blute Convenient Vector Spaces, Convenient Manifolds and Differential Linear Logic Convenient vector spaces II: Definition Theorem Let E be a locally convex vector space. -
Discrete Analytic Convex Geometry Introduction
Discrete Analytic Convex Geometry Introduction Martin Henk Otto-von-Guericke-Universit¨atMagdeburg Winter term 2012/13 webpage CONTENTS i Contents Preface ii 0 Some basic and convex facts1 1 Support and separate5 2 Radon, Helly, Caratheodory and (a few) relatives9 Index 11 ii CONTENTS Preface The material presented here is stolen from different excellent sources: • First of all: A manuscript of Ulrich Betke on convexity which is partially based on lecture notes given by Peter McMullen. • The inspiring books by { Alexander Barvinok, "A course in Convexity" { G¨unter Ewald, "Combinatorial Convexity and Algebraic Geometry" { Peter M. Gruber, "Convex and Discrete Geometry" { Peter M. Gruber and Cerrit G. Lekkerkerker, "Geometry of Num- bers" { Jiri Matousek, "Discrete Geometry" { Rolf Schneider, "Convex Geometry: The Brunn-Minkowski Theory" { G¨unter M. Ziegler, "Lectures on polytopes" • and some original papers !! and they are part of lecture notes on "Discrete and Convex Geometry" jointly written with Maria Hernandez Cifre but not finished yet. Some basic and convex facts 1 0 Some basic and convex facts n 0.1 Notation. R = x = (x1; : : : ; xn)| : xi 2 R denotes the n-dimensional Pn Euclidean space equipped with the Euclidean inner product hx; yi = i=1 xi yi, n p x; y 2 R , and the Euclidean norm jxj = hx; xi. 0.2 Definition [Linear, affine, positive and convex combination]. Let m 2 n N and let xi 2 R , λi 2 R, 1 ≤ i ≤ m. Pm i) i=1 λi xi is called a linear combination of x1;:::; xm. Pm Pm ii) If i=1 λi = 1 then i=1 λi xi is called an affine combination of x1; :::; xm. -
On Modified L 1-Minimization Problems in Compressed Sensing Man Bahadur Basnet Iowa State University
Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2013 On Modified l_1-Minimization Problems in Compressed Sensing Man Bahadur Basnet Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Part of the Applied Mathematics Commons, and the Mathematics Commons Recommended Citation Basnet, Man Bahadur, "On Modified l_1-Minimization Problems in Compressed Sensing" (2013). Graduate Theses and Dissertations. 13473. https://lib.dr.iastate.edu/etd/13473 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. On modified `-one minimization problems in compressed sensing by Man Bahadur Basnet A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Mathematics (Applied Mathematics) Program of Study Committee: Fritz Keinert, Co-major Professor Namrata Vaswani, Co-major Professor Eric Weber Jennifer Davidson Alexander Roitershtein Iowa State University Ames, Iowa 2013 Copyright © Man Bahadur Basnet, 2013. All rights reserved. ii DEDICATION I would like to dedicate this thesis to my parents Jaya and Chandra and to my family (wife Sita, son Manas and daughter Manasi) without whose support, encouragement and love, I would not have been able to complete this work. iii TABLE OF CONTENTS LIST OF TABLES . vi LIST OF FIGURES . vii ACKNOWLEDGEMENTS . viii ABSTRACT . x CHAPTER 1. INTRODUCTION . -
Non-Linear Inner Structure of Topological Vector Spaces
mathematics Article Non-Linear Inner Structure of Topological Vector Spaces Francisco Javier García-Pacheco 1,*,† , Soledad Moreno-Pulido 1,† , Enrique Naranjo-Guerra 1,† and Alberto Sánchez-Alzola 2,† 1 Department of Mathematics, College of Engineering, University of Cadiz, 11519 Puerto Real, CA, Spain; [email protected] (S.M.-P.); [email protected] (E.N.-G.) 2 Department of Statistics and Operation Research, College of Engineering, University of Cadiz, 11519 Puerto Real (CA), Spain; [email protected] * Correspondence: [email protected] † These authors contributed equally to this work. Abstract: Inner structure appeared in the literature of topological vector spaces as a tool to charac- terize the extremal structure of convex sets. For instance, in recent years, inner structure has been used to provide a solution to The Faceless Problem and to characterize the finest locally convex vector topology on a real vector space. This manuscript goes one step further by settling the bases for studying the inner structure of non-convex sets. In first place, we observe that the well behaviour of the extremal structure of convex sets with respect to the inner structure does not transport to non-convex sets in the following sense: it has been already proved that if a face of a convex set intersects the inner points, then the face is the whole convex set; however, in the non-convex setting, we find an example of a non-convex set with a proper extremal subset that intersects the inner points. On the opposite, we prove that if a extremal subset of a non-necessarily convex set intersects the affine internal points, then the extremal subset coincides with the whole set. -
Bootstrapping the Mazur–Orlicz-König Theorem and the Hahn–Banach Lagrange Theorem
Journal of Convex Analysis Volume 25 (2018), No. 2, 1–xx Bootstrapping the Mazur–Orlicz-König Theorem and the Hahn–Banach Lagrange Theorem Stephen Simons Department of Mathematics, University of California, Santa Barbara, CA 93106-3080, U.S.A. [email protected] Dedicated to Antonino Maugeri on the occasion of his 70th birthday. Received: December 25, 2015 Accepted: August 4, 2016 We give some extensions of König’s extension of the Mazur–Orlicz theorem. These extensions include generalizations of a surprising recent result of Sun Chuanfeng, and generalizations to the product of more than two spaces of the “Hahn–Banach–Lagrange” theorem. Keywords: Sublinear functional, convex function, affine function, Hahn–Banach theorem, Mazur–Orlicz–König theorem. 2010 Mathematics Subject Classification: 46A22, 46N10. 1. Introduction In this paper, all vector spaces are real. We shall use the terms sublinear, linear, convex, concave and affine in their usual senses. This paper is about extensions of the Mazur–Orlicz theorem, which first ap- peared in [5]: Let E be a vector space, S : E ! R be sublinear and C be a nonempty convex subset of E. Then there exists a linear map L: E ! R such that L ≤ S on E and infC L = infC S: Early improvements and applications of this result were given, in chronological order, by Sikorski [6], Pták [4], König [1], Landsberg–Schirotzek [3] and König [2]. By a convex–affine version of a known result we mean that it corresponds to the known result with the word sublinear in the hypothesis replaced by convex and the word linear in the conclusion replaced by affine. -
Infinite Dimensional Lie Theory from the Point of View of Functional
Infinite dimensional Lie Theory from the point of view of Functional Analysis Josef Teichmann F¨urmeine Eltern, Michael, Michaela und Hannah Zsofia Hazel. Abstract. Convenient analysis is enlarged by a powerful theory of Hille-Yosida type. More precisely asymptotic spectral properties of bounded operators on a convenient vector space are related to the existence of smooth semigroups in a necessary and sufficient way. An approximation theorem of Trotter-type is proved, too. This approximation theorem is in fact an existence theorem for smooth right evolutions of non-autonomous differential equations on convenient locally convex spaces and crucial for the following applications. To enlighten the generically "unsolved" (even though H. Omori et al. gave interesting and concise conditions for regularity) question of the existence of product integrals on convenient Lie groups, we provide by the given approximation formula some simple criteria. On the one hand linearization is used, on the other hand remarkable families of right invariant distance functions, which exist on all up to now known Lie groups, are the ingredients: Assuming some natural global conditions regularity can be proved on convenient Lie groups. The existence of product integrals is an essential basis for Lie theory in the convenient setting, since generically differential equations cannot be solved on non-normable locally convex spaces. The relationship between infinite dimensional Lie algebras and Lie groups, which is well under- stood in the regular case, is also reviewed from the point of view of local Lie groups: Namely the question under which conditions the existence of a local Lie group for a given convenient Lie algebra implies the existence of a global Lie group is treated by cohomological methods. -
Constructing Sublinear Expectations on Path Space
Constructing Sublinear Expectations on Path Space Marcel Nutz∗ Ramon van Handel† January 21, 2013 Abstract We provide a general construction of time-consistent sublinear ex- pectations on the space of continuous paths. It yields the existence of the conditional G-expectation of a Borel-measurable (rather than quasi-continuous) random variable, a generalization of the random G- expectation, and an optional sampling theorem that holds without exceptional set. Our results also shed light on the inherent limitations to constructing sublinear expectations through aggregation. Keywords Sublinear expectation; G-expectation; random G-expectation; Time- consistency; Optional sampling; Dynamic programming; Analytic set AMS 2000 Subject Classification 93E20; 60H30; 91B30; 28A05 1 Introduction d We study sublinear expectations on the space Ω= C0(R+, R ) of continuous arXiv:1205.2415v3 [math.PR] 11 Apr 2013 paths. Taking the dual point of view, we are interested in mappings P ξ 0(ξ)= sup E [ξ], 7→ E P ∈P where ξ is a random variable and is a set of probability measures, possibly P non-dominated. In fact, any sublinear expectation with certain continuity ∗Dept. of Mathematics, Columbia University, New York. [email protected] The work of MN is partially supported by NSF grant DMS-1208985. †Sherrerd Hall rm. 227, Princeton University, Princeton. [email protected] The work of RvH is partially supported by NSF grant DMS-1005575. 1 properties is of this form (cf. [10, Sect. 4]). Under appropriate assumptions on , we would like to construct a conditional expectation (ξ) at any P Eτ stopping time τ of the the filtration generated by the canonical process {Ft} B and establish the tower property ( (ξ)) = (ξ) for stopping times σ τ, (1.1) Eσ Eτ Eσ ≤ a property also known as time-consistency in this context.