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Uniform Boundedness Principle for Unbounded Operators
UNIFORM BOUNDEDNESS PRINCIPLE FOR UNBOUNDED OPERATORS C. GANESA MOORTHY and CT. RAMASAMY Abstract. A uniform boundedness principle for unbounded operators is derived. A particular case is: Suppose fTigi2I is a family of linear mappings of a Banach space X into a normed space Y such that fTix : i 2 Ig is bounded for each x 2 X; then there exists a dense subset A of the open unit ball in X such that fTix : i 2 I; x 2 Ag is bounded. A closed graph theorem and a bounded inverse theorem are obtained for families of linear mappings as consequences of this principle. Some applications of this principle are also obtained. 1. Introduction There are many forms for uniform boundedness principle. There is no known evidence for this principle for unbounded operators which generalizes classical uniform boundedness principle for bounded operators. The second section presents a uniform boundedness principle for unbounded operators. An application to derive Hellinger-Toeplitz theorem is also obtained in this section. A JJ J I II closed graph theorem and a bounded inverse theorem are obtained for families of linear mappings in the third section as consequences of this principle. Go back Let us assume the following: Every vector space X is over R or C. An α-seminorm (0 < α ≤ 1) is a mapping p: X ! [0; 1) such that p(x + y) ≤ p(x) + p(y), p(ax) ≤ jajαp(x) for all x; y 2 X Full Screen Close Received November 7, 2013. 2010 Mathematics Subject Classification. Primary 46A32, 47L60. Key words and phrases. -
On the Boundary Between Mereology and Topology
On the Boundary Between Mereology and Topology Achille C. Varzi Istituto per la Ricerca Scientifica e Tecnologica, I-38100 Trento, Italy [email protected] (Final version published in Roberto Casati, Barry Smith, and Graham White (eds.), Philosophy and the Cognitive Sciences. Proceedings of the 16th International Wittgenstein Symposium, Vienna, Hölder-Pichler-Tempsky, 1994, pp. 423–442.) 1. Introduction Much recent work aimed at providing a formal ontology for the common-sense world has emphasized the need for a mereological account to be supplemented with topological concepts and principles. There are at least two reasons under- lying this view. The first is truly metaphysical and relates to the task of charac- terizing individual integrity or organic unity: since the notion of connectedness runs afoul of plain mereology, a theory of parts and wholes really needs to in- corporate a topological machinery of some sort. The second reason has been stressed mainly in connection with applications to certain areas of artificial in- telligence, most notably naive physics and qualitative reasoning about space and time: here mereology proves useful to account for certain basic relation- ships among things or events; but one needs topology to account for the fact that, say, two events can be continuous with each other, or that something can be inside, outside, abutting, or surrounding something else. These motivations (at times combined with others, e.g., semantic transpar- ency or computational efficiency) have led to the development of theories in which both mereological and topological notions play a pivotal role. How ex- actly these notions are related, however, and how the underlying principles should interact with one another, is still a rather unexplored issue. -
Baire Category Theorem and Uniform Boundedness Principle)
Functional Analysis (WS 19/20), Problem Set 3 (Baire Category Theorem and Uniform Boundedness Principle) Baire Category Theorem B1. Let (X; k·kX ) be an innite dimensional Banach space. Prove that X has uncountable Hamel basis. Note: This is Problem A2 from Problem Set 1. B2. Consider subset of bounded sequences 1 A = fx 2 l : only nitely many xk are nonzerog : Can one dene a norm on A so that it becomes a Banach space? Consider the same question with the set of polynomials dened on interval [0; 1]. B3. Prove that the set L2(0; 1) has empty interior as the subset of Banach space L1(0; 1). B4. Let f : [0; 1) ! [0; 1) be a continuous function such that for every x 2 [0; 1), f(kx) ! 0 as k ! 1. Prove that f(x) ! 0 as x ! 1. B5.( Uniform Boundedness Principle) Let (X; k · kX ) be a Banach space and (Y; k · kY ) be a normed space. Let fTαgα2A be a family of bounded linear operators between X and Y . Suppose that for any x 2 X, sup kTαxkY < 1: α2A Prove that . supα2A kTαk < 1 Uniform Boundedness Principle U1. Let F be a normed space C[0; 1] with L2(0; 1) norm. Check that the formula 1 Z n 'n(f) = n f(t) dt 0 denes a bounded linear functional on . Verify that for every , F f 2 F supn2N j'n(f)j < 1 but . Why Uniform Boundedness Principle is not satised in this case? supn2N k'nk = 1 U2.( pointwise convergence of operators) Let (X; k · kX ) be a Banach space and (Y; k · kY ) be a normed space. -
FUNCTIONAL ANALYSIS 1. Banach and Hilbert Spaces in What
FUNCTIONAL ANALYSIS PIOTR HAJLASZ 1. Banach and Hilbert spaces In what follows K will denote R of C. Definition. A normed space is a pair (X, k · k), where X is a linear space over K and k · k : X → [0, ∞) is a function, called a norm, such that (1) kx + yk ≤ kxk + kyk for all x, y ∈ X; (2) kαxk = |α|kxk for all x ∈ X and α ∈ K; (3) kxk = 0 if and only if x = 0. Since kx − yk ≤ kx − zk + kz − yk for all x, y, z ∈ X, d(x, y) = kx − yk defines a metric in a normed space. In what follows normed paces will always be regarded as metric spaces with respect to the metric d. A normed space is called a Banach space if it is complete with respect to the metric d. Definition. Let X be a linear space over K (=R or C). The inner product (scalar product) is a function h·, ·i : X × X → K such that (1) hx, xi ≥ 0; (2) hx, xi = 0 if and only if x = 0; (3) hαx, yi = αhx, yi; (4) hx1 + x2, yi = hx1, yi + hx2, yi; (5) hx, yi = hy, xi, for all x, x1, x2, y ∈ X and all α ∈ K. As an obvious corollary we obtain hx, y1 + y2i = hx, y1i + hx, y2i, hx, αyi = αhx, yi , Date: February 12, 2009. 1 2 PIOTR HAJLASZ for all x, y1, y2 ∈ X and α ∈ K. For a space with an inner product we define kxk = phx, xi . Lemma 1.1 (Schwarz inequality). -
The Boundary Manifold of a Complex Line Arrangement
Geometry & Topology Monographs 13 (2008) 105–146 105 The boundary manifold of a complex line arrangement DANIEL CCOHEN ALEXANDER ISUCIU We study the topology of the boundary manifold of a line arrangement in CP2 , with emphasis on the fundamental group G and associated invariants. We determine the Alexander polynomial .G/, and more generally, the twisted Alexander polynomial associated to the abelianization of G and an arbitrary complex representation. We give an explicit description of the unit ball in the Alexander norm, and use it to analyze certain Bieri–Neumann–Strebel invariants of G . From the Alexander polynomial, we also obtain a complete description of the first characteristic variety of G . Comparing this with the corresponding resonance variety of the cohomology ring of G enables us to characterize those arrangements for which the boundary manifold is formal. 32S22; 57M27 For Fred Cohen on the occasion of his sixtieth birthday 1 Introduction 1.1 The boundary manifold Let A be an arrangement of hyperplanes in the complex projective space CPm , m > 1. Denote by V S H the corresponding hypersurface, and by X CPm V its D H A D n complement. Among2 the origins of the topological study of arrangements are seminal results of Arnol’d[2] and Cohen[8], who independently computed the cohomology of the configuration space of n ordered points in C, the complement of the braid arrangement. The cohomology ring of the complement of an arbitrary arrangement A is by now well known. It is isomorphic to the Orlik–Solomon algebra of A, see Orlik and Terao[34] as a general reference. -
A Study of Optimization in Hilbert Space
California State University, San Bernardino CSUSB ScholarWorks Theses Digitization Project John M. Pfau Library 1998 A study of optimization in Hilbert Space John Awunganyi Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd-project Part of the Mathematics Commons Recommended Citation Awunganyi, John, "A study of optimization in Hilbert Space" (1998). Theses Digitization Project. 1459. https://scholarworks.lib.csusb.edu/etd-project/1459 This Project is brought to you for free and open access by the John M. Pfau Library at CSUSB ScholarWorks. It has been accepted for inclusion in Theses Digitization Project by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. A STUDY OF OPTIMIZATIONIN HELBERT SPACE A Project Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment ofthe Requirements for the Degree Master ofArts in Mathematics by John Awunganyi June 1998 A STUDY OF OPTIMIZATION IN HILBERT SPACE A Project Presented to the Faculty of California State University, San Bernardino by John Awunganyi June 1998 Approved by; Dr. Chetan Prakash, Chair, Mathematics Date r. Wenxang Wang Dr. Christopher Freiling Dr. Terry Hallett, Graduate Coordinator Dr. Peter Williams, Department Chair ABSTRACT The primary objective ofthis project is to demonstrate that a certain field ofoptimization can be effectively unified by afew geometric principles ofcomplete normed linear space theory. By employing these principles, important and complex finite - dimensional problems can be interpreted and solved by methods springing from geometric insight. Concepts such as distance, orthogonality, and convexity play a fundamental and indispensable role in this development. Viewed in these terms, seemingly diverse problems and techniques often are found to be closely related. -
Graph Manifolds and Taut Foliations Mark Brittenham1, Ramin Naimi, and Rachel Roberts2 Introduction
Graph manifolds and taut foliations 1 2 Mark Brittenham , Ramin Naimi, and Rachel Roberts UniversityofTexas at Austin Abstract. We examine the existence of foliations without Reeb comp onents, taut 1 1 foliations, and foliations with no S S -leaves, among graph manifolds. We show that each condition is strictly stronger than its predecessors, in the strongest p os- sible sense; there are manifolds admitting foliations of eachtyp e which do not admit foliations of the succeeding typ es. x0 Introduction Taut foliations have b een increasingly useful in understanding the top ology of 3-manifolds, thanks largely to the work of David Gabai [Ga1]. Many 3-manifolds admit taut foliations [Ro1],[De],[Na1], although some do not [Br1],[Cl]. To date, however, there are no adequate necessary or sucient conditions for a manifold to admit a taut foliation. This pap er seeks to add to this confusion. In this pap er we study the existence of taut foliations and various re nements, among graph manifolds. What we show is that there are many graph manifolds which admit foliations that are as re ned as wecho ose, but which do not admit foliations admitting any further re nements. For example, we nd manifolds which admit foliations without Reeb comp onents, but no taut foliations. We also nd 0 manifolds admitting C foliations with no compact leaves, but which do not admit 2 any C such foliations. These results p oint to the subtle nature b ehind b oth top ological and analytical assumptions when dealing with foliations. A principal motivation for this work came from a particularly interesting exam- ple; the manifold M obtained by 37/2 Dehn surgery on the 2,3,7 pretzel knot K . -
A TEXTBOOK of TOPOLOGY Lltld
SEIFERT AND THRELFALL: A TEXTBOOK OF TOPOLOGY lltld SEI FER T: 7'0PO 1.OG 1' 0 I.' 3- Dl M E N SI 0 N A I. FIRERED SPACES This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: SAMUELEILENBERG AND HYMANBASS A list of recent titles in this series appears at the end of this volunie. SEIFERT AND THRELFALL: A TEXTBOOK OF TOPOLOGY H. SEIFERT and W. THRELFALL Translated by Michael A. Goldman und S E I FE R T: TOPOLOGY OF 3-DIMENSIONAL FIBERED SPACES H. SEIFERT Translated by Wolfgang Heil Edited by Joan S. Birman and Julian Eisner @ 1980 ACADEMIC PRESS A Subsidiary of Harcourr Brace Jovanovich, Publishers NEW YORK LONDON TORONTO SYDNEY SAN FRANCISCO COPYRIGHT@ 1980, BY ACADEMICPRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. 11 1 Fifth Avenue, New York. New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NWI 7DX Mit Genehmigung des Verlager B. G. Teubner, Stuttgart, veranstaltete, akin autorisierte englische Ubersetzung, der deutschen Originalausgdbe. Library of Congress Cataloging in Publication Data Seifert, Herbert, 1897- Seifert and Threlfall: A textbook of topology. Seifert: Topology of 3-dimensional fibered spaces. (Pure and applied mathematics, a series of mono- graphs and textbooks ; ) Translation of Lehrbuch der Topologic. Bibliography: p. Includes index. 1. -
General Topology
General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry). -
A Hilbert Space Theory of Generalized Graph Signal Processing
1 A Hilbert Space Theory of Generalized Graph Signal Processing Feng Ji and Wee Peng Tay, Senior Member, IEEE Abstract Graph signal processing (GSP) has become an important tool in many areas such as image pro- cessing, networking learning and analysis of social network data. In this paper, we propose a broader framework that not only encompasses traditional GSP as a special case, but also includes a hybrid framework of graph and classical signal processing over a continuous domain. Our framework relies extensively on concepts and tools from functional analysis to generalize traditional GSP to graph signals in a separable Hilbert space with infinite dimensions. We develop a concept analogous to Fourier transform for generalized GSP and the theory of filtering and sampling such signals. Index Terms Graph signal proceesing, Hilbert space, generalized graph signals, F-transform, filtering, sampling I. INTRODUCTION Since its emergence, the theory and applications of graph signal processing (GSP) have rapidly developed (see for example, [1]–[10]). Traditional GSP theory is essentially based on a change of orthonormal basis in a finite dimensional vector space. Suppose G = (V; E) is a weighted, arXiv:1904.11655v2 [eess.SP] 16 Sep 2019 undirected graph with V the vertex set of size n and E the set of edges. Recall that a graph signal f assigns a complex number to each vertex, and hence f can be regarded as an element of n C , where C is the set of complex numbers. The heart of the theory is a shift operator AG that is usually defined using a property of the graph. -
HILBERT SPACES in MODELLING of SYSTEMS Jean Claude Dutailly
HILBERT SPACES IN MODELLING OF SYSTEMS Jean Claude Dutailly To cite this version: Jean Claude Dutailly. HILBERT SPACES IN MODELLING OF SYSTEMS. 2014. hal-00974251 HAL Id: hal-00974251 https://hal.archives-ouvertes.fr/hal-00974251 Preprint submitted on 5 Apr 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. HILBERT SPACES IN MODELLING OF SYSTEMS Jean Claude Dutailly Paris April 4, 2014 Abstract This paper studies the quantitative models which are used to represent systems, whatever the field (Physics, Economics, Finances,...). We prove that models which meet some very general conditions can be associated to Hilbert vector spaces : the state of the system is then identified with a unique vector. The measure of this state is done through observables, whose properties are studied and we show that they can take only values which are eigen vectors of the observables, and a probability law can be associated to the occurence of these measures. The effect of a change of variables in a model is related to a unitary operator. This leads to general laws for the evolution of systems, and to powerful tools for the study of transitions of phases in a system. -
Homology Groups of Homeomorphic Topological Spaces
An Introduction to Homology Prerna Nadathur August 16, 2007 Abstract This paper explores the basic ideas of simplicial structures that lead to simplicial homology theory, and introduces singular homology in order to demonstrate the equivalence of homology groups of homeomorphic topological spaces. It concludes with a proof of the equivalence of simplicial and singular homology groups. Contents 1 Simplices and Simplicial Complexes 1 2 Homology Groups 2 3 Singular Homology 8 4 Chain Complexes, Exact Sequences, and Relative Homology Groups 9 ∆ 5 The Equivalence of H n and Hn 13 1 Simplices and Simplicial Complexes Definition 1.1. The n-simplex, ∆n, is the simplest geometric figure determined by a collection of n n + 1 points in Euclidean space R . Geometrically, it can be thought of as the complete graph on (n + 1) vertices, which is solid in n dimensions. Figure 1: Some simplices Extrapolating from Figure 1, we see that the 3-simplex is a tetrahedron. Note: The n-simplex is topologically equivalent to Dn, the n-ball. Definition 1.2. An n-face of a simplex is a subset of the set of vertices of the simplex with order n + 1. The faces of an n-simplex with dimension less than n are called its proper faces. 1 Two simplices are said to be properly situated if their intersection is either empty or a face of both simplices (i.e., a simplex itself). By \gluing" (identifying) simplices along entire faces, we get what are known as simplicial complexes. More formally: Definition 1.3. A simplicial complex K is a finite set of simplices satisfying the following condi- tions: 1 For all simplices A 2 K with α a face of A, we have α 2 K.