Functional Analysis and Optimization

Total Page:16

File Type:pdf, Size:1020Kb

Functional Analysis and Optimization Functional Analysis and Optimization Kazufumi Ito∗ November 29, 2016 Abstract In this monograph we develop the function space method for optimization problems and operator equations in Banach spaces. Optimization is the one of key components for mathematical modeling of real world problems and the solution method provides an accurate and essential description and validation of the mathematical model. Op- timization problems are encountered frequently in engineering and sciences and have widespread practical applications.In general we optimize an appropriately chosen cost functional subject to constraints. For example, the constraints are in the form of equal- ity constraints and inequality constraints. The problem is casted in a function space and it is important to formulate the problem in a proper function space framework in order to develop the accurate theory and the effective algorithm. Many of the constraints for the optimization are governed by partial differential and functional equations. In order to discuss the existence of optimizers it is essential to develop a comprehensive treatment of the constraints equations. In general the necessary optimality condition is in the form of operator equations. Non-smooth optimization becomes a very basic modeling toll and enlarges and en- hances the applications of the optimization method in general. For example, in the classical variational formulation we analyze the non Newtonian energy functional and nonsmooth friction penalty. For the imaging/signal analysis the sparsity optimization is used by means of L1 and TV regularization. In order to develop an efficient solu- tion method for large scale optimizations it is essential to develop a set of necessary conditions in the equation form rather than the variational inequality. The Lagrange multiplier theory for the constrained minimizations and nonsmooth optimization prob- lems is developed and analyzed The theory derives the complementarity condition for the multiplier and the solution. Consequently, one can derive the necessary optimality system for the solution and the multiplier for a general class of nonsmooth and noncovex optimizations. In the light of the theory we derive and analyze numerical algorithms based the primal-dual active set method and the semi-smooth Newton method. The monograph also covers the basic materials for real analysis, functional anal- ysis, Banach space theory, convex analysis, operator theory and PDE theory, which makes the book self-contained, comprehensive and complete. The analysis component is naturally connected to the optimization theory. The necessary optimality condition is in general written as nonlinear operator equations for the primal variable and La- grange multiplier. The Lagrange multiplier theory of a general class of nonsmooth and ∗Department of Mathematics, North Carolina State University, Raleigh, North Carolina, USA 1 non-convex optimization are based on the functional analysis tool. For example, the necessary optimality for the constrained optimization is in general nonsmooth and a psuedo-monontone type operator equation. To develop the mathematical treatment of the PDE constrained minimization we also present the PDE theory and linear and nonlinear C0-semigroup theory on Banach spaces and we have a full discussion of well-posedness of the control dynamics and optimization problems. That is, the existence and uniqueness of solutions to a general class of controlled linear and nonlinear PDEs and Cauchy problems for nonlinear evo- lution equations in Banach space are discussed. Especially the linear operator theory and the (psuedo-) monotone operator and mapping theory and the fixed point the- ory.Throughout the monograph demonstrates the theory and algorithm using concrete examples and describes how to apply it for motivated applications 1 Introduction In this monograph we develop the function space approach for the optimization prob- lems, e.g., nonlinear programming in Banach spaces, convex and non-convex nonsmooth variational problems, control and inverse problems, image/signal analysis, material design, classification, and resource allocation. We also develop the basic functional analysis tool for for the nonlinear equations and Cauchy problems in Banach spaces in order to establish the well-posedness of the constraint optimization. The function space optimization method allows us to study the well-posednees for a general class of optimization problems systematically using the functional analysis tool and it also enables us to develop and analyze the numerical optimization algorithm based on the calculus of variations and the Lagrange multiplier theory for the constrained opti- mization. The function space analysis provides the regularizing procedure to remeady the ill-posedness of the constraints and the necessary optimality system and thus one can develop a stable and effective numerical algorithm. The convergence analysis for the function space optimization guides us to analyze and improve the stability and effectiveness of algorithms for the associated large scale optimizations. A general class of constrained optimization problems in function spaces is consid- ered and we develop the Lagrange multiplier theory and effective solution algorithms. Applications are presented and analyzed for various examples including control and design optimization, inverse problems, image analysis and variational problems. Non- convex and non-smooth optimization becomes increasing important for the advanced and effective use of optimization methods and thus they are essential topics for the monograph. We introduce the basic function space and functional analysis tools needed for our analysis. The monograph provides an introduction to the functional analysis, real analysis and convex analysis and their basic concepts with illustrated examples. Thus, one can use the book as a basic course material for the functional analysis and nonlinear operator theory. The nonlinear operator theory and their applications to PDE's problems are presented in details, including classical variational optimization problems in Newtonian and non-Newtonian mechanics and fluid. An emerging appli- cation of optimizations include the imaging and signal analysis and the classification and machine learning. It is often formulated as the PDE's constrained optimization. There are numerous applications of the optimization theory and variational method in all sciences and to real world applications. For example, the followings are the 2 motivated examples. Example 1 (Quadratic programming) 1 t t n min 2 x Ax − a x over x 2 R subject to Ex = b (equality constraint) Gx ≤ c (inequality constraint): where A 2 Rn×n is a symmetric positive matrix, E 2 Rm×n;G 2 Rp×n, and a 2 Rn, b 2 Rm, c 2 Rp are given vectors. The quadratic programming is formulated on a Hilbert space X for the variational problem, in which (Ax; x)X defines the quadratic form on X. Example 2 (Linear programming) The linear programing minimizes (c; x)X subject to the linear constraint Ax = b and x ≥ 0. Its dual problem is to maximize (b; x) subject to A∗x ≤ c. It has many important classes of applications. For example, the optimal transport problem is to find the joint distribution function µ(x; y) ≥ 0 on X × Y that transports the probability density p0(x) to p1(y), i.e. Z Z µ(x; y) dx = p0(x); µ(x; y) dy = p1(y); Y Y with minimum energy Z c(x; y)µ(x; y) dxdy = (c; µ): X×Y Example 3 (Integer programming) Let F is a continuous function Rn. The integer programming is to minimize F (x) over the integer variables x, i.e., min F (x) subject to x 2 f0; 1; ··· Ng: The binary optimizations to minimize F (x) over the binary variables x. The network optimization problem is the one of important applications. Example4 (Obstacle problem) Consider the variational problem; Z 1 1 d min J(u) = ( j u(x)j2 − f(x)u(x)) dx (1.1) 0 2 dx subject to u(x) ≤ (x) 1 over all displacement function u(x) 2 H0 (0; 1), where the function represents the upper bound (obstacle) of the deformation. The functional J represents the Newtonian deformation energy and f(x) is the applied force. Example 5 (Function Interpolation) Consider the interpolation problem; Z 1 2 1 d 2 min ( j 2 u(x)j dx over all functions u(x); (1.2) 0 2 dx subject to the interpolation conditions d u(x ) = b ; u(x ) = c ; 1 ≤ i ≤ m: i i dx i i 3 Example 6 (Image/Signal Analysis) Consider the deconvolution problem of finding the image u that satisfies (Ku)(x) = f(x) for the convolution K Z (Ku)(x) = k(x; y)u(y) dy: Ω over a domain Ω. It is in general ill-posed and we use the (Tikhonov) regularization formulation to have a stable deconvolution solution u, i.e., for Ω = [0; 1] Z 1 α d min (jKu − fj2 + j u(x)j2 + β u(x)) dx over all possible image u ≥ 0; (1.3) 0 2 dx where α; β ≥ 0 are the regularization parameters. The first regularization term is the quadratic varation of image u. Since if u ≥ 0 then juj = u, the regularization term for the L1 sparsity. The two regularization parameters are chosen so that the variation of image u is regularized. Example 7 (L0 optimization) Problem of optimal resource allocation and material dis- tribution and optimal time scheduling can be formulated as Z F (u) + h(u(!)) d! (1.4) Ω α 2 0 with h(u) = 2 juj + β juj where j0j0 = 0; juj0 = 1; otherwise: Since Z ju(!)j0 d! = vol(fu(!) 6= 0g); Ω the solution to (1.4) provides the optimal distribution of u for minimizing the perfor- mance F (u) with appropriate choice of (regularization) parameters α; β > 0. Example 8 (Bayesian statistics) Consider the statical optimization; min −log (p(yjx) + βp(x)) over all admissible parameters x; (1.5) where p(yjx) is the conditional probability density of observation y given parameter x and the p(x) is the prior probability density for parameter x.
Recommended publications
  • Quelques Contributions À La Théorie De La Commande Optimale Déterministe Et Stochastique
    Université de Limoges Habilitation à diriger des recherches présentée par Francisco J. Silva Spécialité: Mathématiques Appliquées Quelques contributions à la théorie de la commande optimale déterministe et stochastique. Mémoire présenté le 15 novembre 2016 devant le jury composé de : Yves Achdou Rapporteur Guillaume Carlier Rapporteur Eduardo Casas Rapporteur Frédéric Bonnans Examinateur Pierre Cardaliaguet Examinateur Alejandro Jofré Examinateur Filippo Santambrogio Examinateur Lionel Thibault Examinateur Samir Adly Chargé de suivi ii A Fernanda. 1 Remerciements Je tiens tout d’abord à remercier Samir Adly pour avoir généreusement ac- cepté d’être le chargé de suivi de ce mémoire. Je voudrais remercier sincérement les rapporteurs de ce mémoire: Yves Achdou, dont les travaux sur l’approximation numérique des jeux à champ moyen sont une réference dans le domaine; Guillaume Carlier, pour l’intêret qu’il a témoigné pour mes travaux; et Eduardo Casas, dont les résultats sur les conditions d’optimalité en contrôle optimale des EDPs ont inspiré ma recherche dans ce domaine. J’adresse mes remerciements aussi aux autres membres du jury: Frédéric Bonnans, mon directeur de thèse qui a toujours suivi de près mes recherches et qui m’a soutenu depuis de longues années; Pierre Cardaliaguet, qui est une réference dans les domaines de jeux différentielles et de jeux à champ moyen; Alejandro Jofré, pour son accueil toujours chaleureux lors de plusieurs séjours de recherche au CMM, Université du Chili; Filippo Santambrogio, pour son intêret pour mes travaux en jeux à champs moyen et pour m’avoir proposé généreusement de travailler avec son ancien étudiant Alpar Richárd Mészáros sur un projet dont l’idée originale lui appartenait; et Lionel Thibault, pour me faire l’honneur d’être membre de mon jury.
    [Show full text]
  • Fundamentals of Functional Analysis Kluwer Texts in the Mathematical Sciences
    Fundamentals of Functional Analysis Kluwer Texts in the Mathematical Sciences VOLUME 12 A Graduate-Level Book Series The titles published in this series are listed at the end of this volume. Fundamentals of Functional Analysis by S. S. Kutateladze Sobolev Institute ofMathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia Springer-Science+Business Media, B.V. A C.I.P. Catalogue record for this book is available from the Library of Congress ISBN 978-90-481-4661-1 ISBN 978-94-015-8755-6 (eBook) DOI 10.1007/978-94-015-8755-6 Translated from OCHOBbI Ij)YHK~HOHaJIl>HODO aHaJIHsa. J/IS;l\~ 2, ;l\OIIOJIHeHHoe., Sobo1ev Institute of Mathematics, Novosibirsk, © 1995 S. S. Kutate1adze Printed on acid-free paper All Rights Reserved © 1996 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1996. Softcover reprint of the hardcover 1st edition 1996 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Contents Preface to the English Translation ix Preface to the First Russian Edition x Preface to the Second Russian Edition xii Chapter 1. An Excursion into Set Theory 1.1. Correspondences . 1 1.2. Ordered Sets . 3 1.3. Filters . 6 Exercises . 8 Chapter 2. Vector Spaces 2.1. Spaces and Subspaces ... ......................... 10 2.2. Linear Operators . 12 2.3. Equations in Operators ........................ .. 15 Exercises . 18 Chapter 3. Convex Analysis 3.1.
    [Show full text]
  • Locally Convex Spaces for Which Λ(E) = Λ[E] and the Dvoretsky-Rogers Theorem Compositio Mathematica, Tome 35, No 2 (1977), P
    COMPOSITIO MATHEMATICA N. DE GRANDE-DE KIMPE Locally convex spaces for which L(E) = L[E] and the Dvoretsky-Rogers theorem Compositio Mathematica, tome 35, no 2 (1977), p. 139-145 <http://www.numdam.org/item?id=CM_1977__35_2_139_0> © Foundation Compositio Mathematica, 1977, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infrac- tion 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/ COMPOSITIO MATHEMATICA, Vol. 35, Fasc. 2, 1977, pag. 139-145 Noordhoff International Publishing Printed in the Netherlands LOCALLY CONVEX SPACES FOR WHICH 039B(E) = 039B [E] AND THE DVORETSKY-ROGERS THEOREM N. De Grande-De Kimpe The classical Dvoretsky-Rogers theorem states that if E is a Banach space for which l1[E] = (l(E), then E is finite dimensional. This property still holds for any (P (1 p ~) (see [5]). Recently it has been shown (see [11]) that the result remains true when one replaces e’ by any non-nuclear perfect sequence space A, having the normal topology n(A, Ax). (This situation does not contain the (P (1 p -)-case). The question whether the Dvoretsky-Rogers theorem holds for any perfect Banach sequence space A is still open. (A partial, positive answer to this problem, generalizing the é’-case for any p is given in this paper.) It seemed however more convenient to us to tackle the problem from the "locally-convex view-point".
    [Show full text]
  • The Tensor Product Representation of Polynomials of Weak Type in a DF-Space
    Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 795016, 7 pages http://dx.doi.org/10.1155/2014/795016 Research Article The Tensor Product Representation of Polynomials of Weak Type in a DF-Space Masaru Nishihara1 and Kwang Ho Shon2 1 Department of Computer Science and Engineering, Faculty of Information Engineering, Fukuoka Institute of Technology, Fukuoka 811-0295, Japan 2 Department of Mathematics, College of Natural Sciences, Pusan National University, Busan 609-735, Republic of Korea Correspondence should be addressed to Kwang Ho Shon; [email protected] Received 15 December 2013; Accepted 19 February 2014; Published 27 March 2014 Academic Editor: Zong-Xuan Chen Copyright © 2014 M. Nishihara and K. H. Shon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let and be locally convex spaces over C and let ( ; ) be the space of all continuous -homogeneous polynomials from to .Wedenoteby⨂,, the -fold symmetric tensor product space of endowed with the projective topology. Then, it is well known that each polynomial ∈(; ) is represented as an element in the space (⨂,,; ) of all continuous linear mappings from ⨂,, to . A polynomial ∈(; ) is said to be of weak type if, for every bounded set of , | is weakly continuous on .Wedenoteby( ; ) the space of all -homogeneous polynomials of weak type from to . In this paper, in case that is a DF space, we will give the tensor product representation of the space ( ; .) 1. Notations and Preliminaries by B() thefamilyofallboundedsubsetsof.
    [Show full text]
  • Document Downloaded From: This Paper Must Be Cited As: the Final Publication Is Available at Copyright
    Document downloaded from: http://hdl.handle.net/10251/43075 This paper must be cited as: Bonet Solves, JA.; Albanese, AA.; Ricker, WJ. (2013). Convergence of arithmetic means of operators in Fréchet spaces. Journal of Mathematical Analysis and Applications. 401(1):160-173. doi:10.1016/j.jmaa.2012.11.060. The final publication is available at http://dx.doi.org/10.1016/j.jmaa.2012.11.060 Copyright Elsevier CONVERGENCE OF ARITHMETIC MEANS OF OPERATORS IN FRÉCHET SPACES ANGELA A. ALBANESE, JOSÉ BONET* AND WERNER J. RICKER Authors' addresses: Angela A. Albanese: Dipartimento di Matematica e Fisica E. De Giorgi, Università del Salento- C.P.193, I-73100 Lecce, Italy email: [email protected] José Bonet (corresponding author): Instituto Universitario de Matemática Pura y Aplicada IUMPA, Universitat Politècnica de València, E-46071 Valencia, Spain email:[email protected] telephone number: +34963879497. fax number: +34963879494. Werner J. Ricker: Math.-Geogr. Fakultät, Katholische Universität Eichstätt- Ingolstadt, D-85072 Eichstätt, Germany email:[email protected] Abstract. Every Köthe echelon Fréchet space X that is Montel and not isomorphic to a countable product of copies of the scalar eld admits a power bounded continuous linear operator T such that I ¡ T does not have closed range, but the sequence of arithmetic means of the iterates of T converge to 0 uniformly on the bounded sets in X. On the other hand, if X is a Fréchet space which does not have a quotient isomorphic to a nuclear Köthe echelon space with a continuous norm, then the sequence of arithmetic means of the iterates of any continuous linear operator T (for which (1=n)T n converges to 0 on the bounded sets) converges uniformly on the bounded subsets of X, i.e., T is uniformly mean ergodic, if and only if the range of I ¡ T is closed.
    [Show full text]
  • Functional Analysis 1 Winter Semester 2013-14
    Functional analysis 1 Winter semester 2013-14 1. Topological vector spaces Basic notions. Notation. (a) The symbol F stands for the set of all reals or for the set of all complex numbers. (b) Let (X; τ) be a topological space and x 2 X. An open set G containing x is called neigh- borhood of x. We denote τ(x) = fG 2 τ; x 2 Gg. Definition. Suppose that τ is a topology on a vector space X over F such that • (X; τ) is T1, i.e., fxg is a closed set for every x 2 X, and • the vector space operations are continuous with respect to τ, i.e., +: X × X ! X and ·: F × X ! X are continuous. Under these conditions, τ is said to be a vector topology on X and (X; +; ·; τ) is a topological vector space (TVS). Remark. Let X be a TVS. (a) For every a 2 X the mapping x 7! x + a is a homeomorphism of X onto X. (b) For every λ 2 F n f0g the mapping x 7! λx is a homeomorphism of X onto X. Definition. Let X be a vector space over F. We say that A ⊂ X is • balanced if for every α 2 F, jαj ≤ 1, we have αA ⊂ A, • absorbing if for every x 2 X there exists t 2 R; t > 0; such that x 2 tA, • symmetric if A = −A. Definition. Let X be a TVS and A ⊂ X. We say that A is bounded if for every V 2 τ(0) there exists s > 0 such that for every t > s we have A ⊂ tV .
    [Show full text]
  • HYPERCYCLIC SUBSPACES in FRÉCHET SPACES 1. Introduction
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 134, Number 7, Pages 1955–1961 S 0002-9939(05)08242-0 Article electronically published on December 16, 2005 HYPERCYCLIC SUBSPACES IN FRECHET´ SPACES L. BERNAL-GONZALEZ´ (Communicated by N. Tomczak-Jaegermann) Dedicated to the memory of Professor Miguel de Guzm´an, who died in April 2004 Abstract. In this note, we show that every infinite-dimensional separable Fr´echet space admitting a continuous norm supports an operator for which there is an infinite-dimensional closed subspace consisting, except for zero, of hypercyclic vectors. The family of such operators is even dense in the space of bounded operators when endowed with the strong operator topology. This completes the earlier work of several authors. 1. Introduction and notation Throughout this paper, the following standard notation will be used: N is the set of positive integers, R is the real line, and C is the complex plane. The symbols (mk), (nk) will stand for strictly increasing sequences in N.IfX, Y are (Hausdorff) topological vector spaces (TVSs) over the same field K = R or C,thenL(X, Y ) will denote the space of continuous linear mappings from X into Y , while L(X)isthe class of operators on X,thatis,L(X)=L(X, X). The strong operator topology (SOT) in L(X) is the one where the convergence is defined as pointwise convergence at every x ∈ X. A sequence (Tn) ⊂ L(X, Y )issaidtobeuniversal or hypercyclic provided there exists some vector x0 ∈ X—called hypercyclic for the sequence (Tn)—such that its orbit {Tnx0 : n ∈ N} under (Tn)isdenseinY .
    [Show full text]
  • 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.
    [Show full text]
  • Stability for Periodic Evolution Families of Bounded Linear Operators
    Surveys in Mathematics and its Applications ISSN 1842-6298 (electronic), 1843-7265 (print) Volume 10 (2015), 61 { 93 STABILITY FOR PERIODIC EVOLUTION FAMILIES OF BOUNDED LINEAR OPERATORS Olivia Saierli Abstract The long time behavior for solutions of evolution periodic equations are reviewed. 1 Introduction The study of differential autonomous systems, especially when the "coefficients" are unbounded (the case of infinitesimal generators, for example) is well represented in the contemporary mathematics literature. The study of asymptotic behavior of solutions of systems with time-varying coefficients is more difficult since, inthis context, the well known spectral criteria (such as those existing in the autonomous case) are no longer valid. In the following, we describe the history of the issue and also we state new results concerning asymptotic behavior of solutions of non- autonomous periodic systems. Let A be a bounded linear operator acting on a Banach space X, x an arbitrary vector in X and let µ be a real parameter. First consider the autonomous system y_(t) = Ay(t); t ≥ 0 (1.1) and the associated inhomogeneous Cauchy Problem { y_(t) = Ay(t) + eiµtx; t ≥ 0 (1.2) y(0) = 0: It is well-known, [30], [6], that the system (1.1) is uniformly exponentially stable (i.e. there exist two positive constants N and ν such that ketAk ≤ Ne−νt) if and 2010 Mathematics Subject Classification: 42A16; 45A05; 47A10; 47A35; 47D06; 47G10; 93D20. Keywords: Periodic evolution families; uniform exponential stability; boundedness; evolution semigroup; almost periodic functions. This work is based on the author's PhD thesis. ****************************************************************************** http://www.utgjiu.ro/math/sma 62 O.
    [Show full text]
  • A Characterisation of C*-Algebras 449
    Proceedings of the Edinburgh Mathematical Society (1987) 30, 445-4S3 © A CHARACTERISATION OF C-ALGEBRAS by M. A. HENNINGS (Received 24th March 1986, revised 10th June 1986) 1. Introduction It is of some interest to the theory of locally convex "-algebras to know under what conditions such an algebra A is a pre-C*-algebra (the topology of A can be described by a submultiplicative norm such that ||x*x|| = ||x||2,Vx6/l). We recall that a locally convex ""-algebra is a complex *-algebra A with the structure of a Hausdorff locally convex topological vector space such that the multiplication is separately continuous, and the involution is continuous. Allan [3] has studied the problem for normed algebras, showing that a unital Banach •-algebra is a C*-algebra if and only if A is symmetric and the set 3B of all absolutely convex hermitian idempotent closed bounded subsets of A has a maximal element. Recalling that an element x of a locally convex algebra A is bounded if there exists A > 0 such that the set {(lx)":neN} is bounded, we find the following simple generalisation of Allan's result: Proposition 0. A unital locally convex *'-algebra A is a C'-algebra if and only if: (a) A is sequentially complete; (b) A is barreled; (c) every element of A is bounded; (d) A is symmetric; (e) the family 38 of absolutely convex hermitian idempotent closed bounded subsets of A has a maximal element Bo. Proof. For any x = x* in A, we can find X > 0 such that the closed absolutely convex hull of the set {(/bc)":ne^J} is contained in Bo.
    [Show full text]
  • Saturating Constructions for Normed Spaces II
    Saturating Constructions for Normed Spaces II Stanislaw J. Szarek∗ Nicole Tomczak-Jaegermann† Abstract We prove several results of the following type: given finite dimensional normed space V possessing certain geometric property there exists another space X hav- ing the same property and such that (1) log dim X = O(log dim V ) and (2) ev- ery subspace of X, whose dimension is not “too small,” contains a further well- complemented subspace nearly isometric to V . This sheds new light on the struc- ture of large subspaces or quotients of normed spaces (resp., large sections or lin- ear images of convex bodies) and provides definitive solutions to several problems stated in the 1980s by V. Milman. 1 Introduction This paper continues the study of the saturation phenomenon that was dis- covered in [ST] and of the effect it has on our understanding of the structure of high-dimensional normed spaces and convex bodies. In particular, we ob- tain here a dichotomy-type result which offers a near definitive treatment of some aspects of the phenomenon. We sketch first some background ideas and hint on the broader motivation explaining the interest in the subject. Much of geometric functional analysis revolves around the study of the family of subspaces (or, dually, of quotients) of a given Banach space. In the finite dimensional case this has a clear geometric interpretation: a normed space is determined by its unit ball, a centrally symmetric convex body, subspaces correspond to sections of that body, and quotients to projections (or, more generally, linear images). Such considerations are very natural from the geometric or linear-algebraic point of view, but they also have a ∗Supported in part by a grant from the National Science Foundation (U.S.A.).
    [Show full text]
  • NONARCHIMEDEAN COALGEBRAS and COADMISSIBLE MODULES 2 of Y
    NONARCHIMEDEAN COALGEBRAS AND COADMISSIBLE MODULES ANTON LYUBININ Abstract. We show that basic notions of locally analytic representation the- ory can be reformulated in the language of topological coalgebras (Hopf alge- bras) and comodules. We introduce the notion of admissible comodule and show that it corresponds to the notion of admissible representation in the case of compact p-adic group. Contents Introduction 1 1. Banach coalgebras 4 1.1. Banach -Coalgebras 5 ̂ 1.2. Constructions⊗ in the category of Banach -coalgebras 6 ̂ 1.3. Banach -bialgebras and Hopf -algebras⊗ 8 ̂ ̂ 1.4. Constructions⊗ in the category of⊗ Banach -bialgebras and Hopf ̂ -algebras. ⊗ 9 ̂ 2. Banach comodules⊗ 9 2.1. Basic definitions 9 2.2. Constructions in the category of Banach -comodules 10 ̂ 2.3. Induction ⊗ 11 2.4. Rational -modules 14 ̂ 2.5. Tensor identities⊗ 15 3. Locally convex -coalgebras 16 ̂ Preliminaries ⊗ 16 3.1. Topological Coalgebras 18 3.2. Topological Bialgebras and Hopf algebras. 20 4. modules and comodules 21 arXiv:1410.3731v2 [math.RA] 26 Jul 2017 4.1. Definitions 21 4.2. Rationality 22 4.3. Quotients, subobjects and simplicity 22 4.4. Cotensor product 23 5. Admissibility 24 Appendix 28 References 29 Introduction The study of p-adic locally analytic representation theory of p-adic groups seems to start in 1980s, with the first examples of such representations studied in the works 1 NONARCHIMEDEAN COALGEBRAS AND COADMISSIBLE MODULES 2 of Y. Morita [M1, M2, M3] (and A. Robert, around the same time), who considered locally analytic principal series representations for p-adic SL2.
    [Show full text]