An Introduction to Functional Analysis
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Constructive Closed Range and Open Mapping Theorems in This Paper
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Indag. Mathem., N.S., 11 (4), 509-516 December l&2000 Constructive Closed Range and Open Mapping Theorems by Douglas Bridges and Hajime lshihara Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand email: d. [email protected] School of Information Science, Japan Advanced Institute of Science and Technology, Tatsunokuchi, Ishikawa 923-12, Japan email: [email protected] Communicated by Prof. AS. Troelstra at the meeting of November 27,200O ABSTRACT We prove a version of the closed range theorem within Bishop’s constructive mathematics. This is applied to show that ifan operator Ton a Hilbert space has an adjoint and a complete range, then both T and T’ are sequentially open. 1. INTRODUCTION In this paper we continue our constructive exploration of the theory of opera- tors, in particular operators on a Hilbert space ([5], [6], [7]). We work entirely within Bishop’s constructive mathematics, which we regard as mathematics with intuitionistic logic. For discussions of the merits of this approach to mathematics - in particular, the multiplicity of its models - see [3] and [ll]. The technical background needed in our paper is found in [I] and [IO]. Our main aim is to prove the following result, the constructive Closed Range Theorem for operators on a Hilbert space (cf. [18], pages 99-103): Theorem 1. Let H be a Hilbert space, and T a linear operator on H such that T* exists and ran(T) is closed. -
Operator Algebras Generated by Left Invertibles Derek Desantis [email protected]
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Dissertations, Theses, and Student Research Papers Mathematics, Department of in Mathematics 3-2019 Operator algebras generated by left invertibles Derek DeSantis [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/mathstudent Part of the Analysis Commons, Harmonic Analysis and Representation Commons, and the Other Mathematics Commons DeSantis, Derek, "Operator algebras generated by left invertibles" (2019). Dissertations, Theses, and Student Research Papers in Mathematics. 93. https://digitalcommons.unl.edu/mathstudent/93 This Article is brought to you for free and open access by the Mathematics, Department of at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Dissertations, Theses, and Student Research Papers in Mathematics by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. OPERATOR ALGEBRAS GENERATED BY LEFT INVERTIBLES by Derek DeSantis A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfilment of Requirements For the Degree of Doctor of Philosophy Major: Mathematics Under the Supervision of Professor David Pitts Lincoln, Nebraska March, 2019 OPERATOR ALGEBRAS GENERATED BY LEFT INVERTIBLES Derek DeSantis, Ph.D. University of Nebraska, 2019 Adviser: David Pitts Operator algebras generated by partial isometries and their adjoints form the basis for some of the most well studied classes of C*-algebras. Representations of such algebras encode the dynamics of orthonormal sets in a Hilbert space. We instigate a research program on concrete operator algebras that model the dynamics of Hilbert space frames. The primary object of this thesis is the norm-closed operator algebra generated by a left y invertible T together with its Moore-Penrose inverse T . -
International Centre for Theoretical Physics
zrrl^^^'C^z ic/89/30i INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS ON THE APPROXIMATIVE NORMAL VALUES OF MULTIVALUES OPERATORS IN TOPOLOGICAL VECTOR SPACE Nguyen Minh Chuong and INTERNATIONAL ATOMIC ENERGY Khuat van Ninh AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION IC/89/301 SUNTO International Atomic Energy Agency and In questo articolo si considera il problemadell'approssimaztone di valori normali di oper- United Nations Educational Scientific and Cultural Organization atori chiusi lineari multivoci dallo spazio vettoriale topologico de Mackey, a valori in un E-spazio. INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS Si dimostrano l'esistenza di un valore normale e la convergenza dei valori approssimanti il valore normale. ON THE APPROXIMATIVE NORMAL VALUES OF MULTIVALUED OPERATORS IN TOPOLOGICAL VECTOR SPACE * Nguyen Minn Chuong * International Centre for Theoretical Physics, Trieste, Italy. and Khuat van Ninh Department of Numerical Analysis, Institute of Mathematics, P.O. Box 631, Bo Ho, 10 000 Hanoi, Vietnam. ABSTRACT In this paper the problem of approximation of normal values of multivalued linear closed operators from topological vector Mackey space into E-space is considered. Existence of normal value and covergence of approximative values to normal value are proved. MIRAMARE - TRIESTE September 1989 To be submitted for publication. Permanent address: Department of Numerical Analysis, Institute of Mathematics, P.O. Box 631, Bo Ho 10 000 Hanoi, Vietnam. 1. Introduction Let X , Y be closed Eubspaces of X,Y respectively, where X is normed In recent years problems on normal value and on approximation of space and Y is Hackey space. Let T be a linear multivalued operator from normal value of multivalued linear operators have been investigated Y to X . -
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. -
An Introduction to Basic Functional Analysis
AN INTRODUCTION TO BASIC FUNCTIONAL ANALYSIS T. S. S. R. K. RAO . 1. Introduction This is a write-up of the lectures given at the Advanced Instructional School (AIS) on ` Functional Analysis and Harmonic Analysis ' during the ¯rst week of July 2006. All of the material is standard and can be found in many basic text books on Functional Analysis. As the prerequisites, I take the liberty of assuming 1) Basic Metric space theory, including completeness and the Baire category theorem. 2) Basic Lebesgue measure theory and some abstract (σ-¯nite) measure theory in- cluding the Radon-Nikodym theorem and the completeness of Lp-spaces. I thank Professor A. Mangasuli for carefully proof reading these notes. 2. Banach spaces and Examples. Let X be a vector space over the real or complex scalar ¯eld. Let k:k : X ! R+ be a function such that (1) kxk = 0 , x = 0 (2) k®xk = j®jkxk (3) kx + yk · kxk + kyk for all x; y 2 X and scalars ®. Such a function is called a norm on X and (X; k:k) is called a normed linear space. Most often we will be working with only one speci¯c k:k on any given vector space X thus we omit writing k:k and simply say that X is a normed linear space. It is an easy exercise to show that d(x; y) = kx ¡ yk de¯nes a metric on X and thus there is an associated notion of topology and convergence. If X is a normed linear space and Y ½ X is a subspace then by restricting the norm to Y , we can consider Y as a normed linear space. -
On Bornivorous Set
On Bornivorous Set By Fatima Kamil Majeed Al-Basri University of Al-Qadisiyah College Of Education Department of Mathematics E-mail:[email protected] Abstract :In this paper, we introduce the concept of the bornivorous set and its properties to construct bornological topological space .Also, we introduce and study the properties related to this concepts like bornological base, bornological subbase , bornological closure set, bornological interior set, bornological frontier set and bornological subspace . Key words : bornivorous set , bornological topological space,b-open set 1.Introduction- The space of entire functions over the complex field C was introduced by Patwardhan who defined a metric on this space by introducing a real-valued map on it[6]. In(1971), H.Hogbe- Nlend introduced the concepts of bornology on a set [3].Many workers such as Dierolf and Domanski, Jan Haluska and others had studied various bornological properties[2]. In this paper at the second section ,bornivorous set has been introduced with some related concepts. While in the third section a new space “Bornological topological space“ has been defined and created in the base of bornivorous set . The bornological topological space also has been explored and its properties .The study also extended to the concepts of the bornological base and bornological subbase of bornological topological space .In the last section a new concepts like bornological closure set, bornological drived set, bornological dense set, bornological interior set, bornological exterior set, bornological frontier set and bornological topological subspace, have been studied with supplementary properties and results which related to them. 1 Definition1.1[3] Let A and B be two subsets of a vector space E. -
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. -
Mackey +0-Barrelled Spaces Stephen A
Advances in Mathematics 145, 230238 (1999) Article ID aima.1998.1815, available online at http:ÂÂwww.idealibrary.com on Mackey +0-Barrelled Spaces Stephen A. Saxon* Department of Mathematics, University of Florida, P.O. Box 118105, Gainesville, Florida 32611-8105 E-mail: saxonÄmath.ufl.edu and Ian Tweddle* Department of Mathematics, University of Strathclyde, Glasgow G11XH, Scotland, United Kingdom E-mail: i.tweddleÄstrath.ac.uk CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - PublisherReceived Connector March 25, 1998; accepted December 14, 1998 In the context of ``Reinventing weak barrelledness,'' the best possible versions of the RobertsonSaxonRobertson Splitting Theorem and the SaxonTweddle Fit and Flat Components Theorem are obtained by weakening the hypothesis from ``barrelled'' to ``Mackey and +0-barreled.'' An example showing that the latter does not imply the former validates novelty, answers an old question, and completes a robust linear picture of ``Mackey weak barrelledness'' begun several decades ago. 1999 Academic Press Key Words: weak barrelledness; Mackey topology; splitting theorem. 1. INTRODUCTION Barrelled spaces have occupied an important place in the theory of locally convex spaces since its earliest days. This is probably due to the fact that they provide a vehicle for generalizing certain important properties of Banach spaces; for example, they form the class of domain spaces for a natural generalization of Banach's closed graph theorem and, in their dual characterization, they embody the conclusion of the BanachSteinhaus theorem for continuous linear functionals. Unlike Banach spaces, they are closed under the taking of inductive limits, countable-codimensional subspaces, products, etc. A barrel in a locally convex space is a closed * We are grateful for support and hospitality from EPSRC GRÂL67257 and Strathclyde University, and for helpful discussions with Professors J. -
Closed Graph Theorems for Bornological Spaces
Khayyam J. Math. 2 (2016), no. 1, 81{111 CLOSED GRAPH THEOREMS FOR BORNOLOGICAL SPACES FEDERICO BAMBOZZI1 Communicated by A. Peralta Abstract. The aim of this paper is that of discussing closed graph theorems for bornological vector spaces in a self-contained way, hoping to make the subject more accessible to non-experts. We will see how to easily adapt classical arguments of functional analysis over R and C to deduce closed graph theorems for bornological vector spaces over any complete, non-trivially valued field, hence encompassing the non-Archimedean case too. We will end this survey by discussing some applications. In particular, we will prove De Wilde's Theorem for non-Archimedean locally convex spaces and then deduce some results about the automatic boundedness of algebra morphisms for a class of bornological algebras of interest in analytic geometry, both Archimedean (complex analytic geometry) and non-Archimedean. Introduction This paper aims to discuss the closed graph theorems for bornological vector spaces in a self-contained exposition and to fill a gap in the literature about the non-Archimedean side of the theory at the same time. In functional analysis over R or C bornological vector spaces have been used since a long time ago, without becoming a mainstream tool. It is probably for this reason that bornological vector spaces over non-Archimedean valued fields were rarely considered. Over the last years, for several reasons, bornological vector spaces have drawn new attentions: see for example [1], [2], [3], [5], [15] and [22]. These new developments involve the non-Archimedean side of the theory too and that is why it needs adequate foundations. -
Quasi-Barrelled Locally Convex Spaces 811
i960] quasi-barrelled locally convex spaces 811 Bibliography 1. R. E. Lane, Absolute convergence of continued fractions, Proc. Amer. Math. Soc. vol. 3 (1952) pp. 904-913. 2. R. E. Lane and H. S. Wall, Continued fractions with absolutely convergent even and odd parts, Trans. Amer. Math. Soc. vol. 67 (1949) pp. 368-380. 3. W. T. Scott and H. S. Wall, A convergence theorem for continued fractions, Trans. Amer. Math. Soc. vol. 47 (1940) pp. 155-172. 4. H. S. Wall, Analytic theory of continued fractions, New York, D. Van Nostrand Company, Inc., 1948. The University of Texas and University of Houston QUASI-BARRELLED LOCALLY CONVEX SPACES MARK MAHOWALD AND GERALD GOULD 1. Introduction and preliminary definitions. The main object of this paper is to answer some problems posed by Dieudonné in his paper Denumerability conditions in locally convex vector spaces [l]. His two main results are as follows: Proposition 1. If Eis a barrelled space on which there is a countable fundamental system of convex compact subsets, [Definition 1.2] then it is the strong dual of a Fréchet-Montel Space. Proposition 2. If E is either bornological or barrelled, and if there is a countable fundamental system of compact subsets, then E is dense in the strong dual of a Fréchet-Montel Space. Two questions raised by Dieudonné in connection with these results are: (a) If E is either bornological or barrelled then it is certainly quasi- barrelled [l, Chapter 3, §2, Example 12]. Can one substitute this weaker condition on E in Proposition 2? (b) Is there is an example of a quasi-barrelled space which is neither barrelled nor bornological? We shall show that the answer to (a) is "Yes," and that the answer to (b) is also "Yes," so that the generalization is in fact a real one. -
Frames and Representing Systems in Fréchet Spaces and Their Duals
Frames and representing systems in Fr´echet spaces and their duals J. Bonet, C. Fern´andez,A. Galbis, J.M. Ribera Abstract Frames and Bessel sequences in Fr´echet spaces and their duals are defined and studied. Their relation with Schauder frames and representing systems is analyzed. The abstract results presented here, when applied to concrete spaces of analytic functions, give many examples and consequences about sampling sets and Dirichlet series expansions. 1 Introduction and preliminaries The purpose of this paper is twofold. On the one hand we study Λ-Bessel sequences 0 (gi)i ⊂ E , Λ-frames and frames with respect to Λ in the dual of a Hausdorff locally convex space E, in particular for Fr´echet spaces and complete (LB)-spaces E, with Λ a sequence space. We investigate the relation of these concepts with representing systems in the sense of Kadets and Korobeinik [12] and with the Schauder frames, that were investigated by the authors in [7]. On the other hand our article emphasizes the deep connection of frames for Fr´echet and (LB)-spaces with the sufficient and weakly sufficient sets for weighted Fr´echet and (LB)-spaces of holomorphic functions. These concepts correspond to sampling sets in the case of Banach spaces of holomorphic functions. Our general results in Sections 2 and 3 permit us to obtain as a consequence many examples and results in the literature in a unified way in Section 4. Section 2 of our article is inspired by the work of Casazza, Christensen and Stoeva [9] in the context of Banach spaces. -
Methods of Applied Mathematics
Methods of Applied Mathematics Todd Arbogast and Jerry L. Bona Department of Mathematics, and Institute for Computational Engineering and Sciences The University of Texas at Austin Copyright 1999{2001, 2004{2005, 2007{2008 (corrected version) by T. Arbogast and J. Bona. Contents Chapter 1. Preliminaries 5 1.1. Elementary Topology 5 1.2. Lebesgue Measure and Integration 13 1.3. Exercises 23 Chapter 2. Normed Linear Spaces and Banach Spaces 27 2.1. Basic Concepts and Definitions. 27 2.2. Some Important Examples 34 2.3. Hahn-Banach Theorems 43 2.4. Applications of Hahn-Banach 48 2.5. The Embedding of X into its Double Dual X∗∗ 52 2.6. The Open Mapping Theorem 53 2.7. Uniform Boundedness Principle 57 2.8. Compactness and Weak Convergence in a NLS 58 2.9. The Dual of an Operator 63 2.10. Exercises 66 Chapter 3. Hilbert Spaces 73 3.1. Basic Properties of Inner-Products 73 3.2. Best Approximation and Orthogonal Projections 75 3.3. The Dual Space 78 3.4. Orthonormal Subsets 79 3.5. Weak Convergence in a Hilbert Space 86 3.6. Exercises 87 Chapter 4. Spectral Theory and Compact Operators 89 4.1. Definitions of the Resolvent and Spectrum 90 4.2. Basic Spectral Theory in Banach Spaces 91 4.3. Compact Operators on a Banach Space 93 4.4. Bounded Self-Adjoint Linear Operators on a Hilbert Space 99 4.5. Compact Self-Adjoint Operators on a Hilbert Space 104 4.6. The Ascoli-Arzel`aTheorem 107 4.7. Sturm Liouville Theory 109 4.8.