DOCSLIB.ORG

Elements of Topology and Functional Analysis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Elements of Topology and Functional Analysis Appendix A Elements of topology and functional analysis Here, for the reader’s convenience, we collect fundamental concepts, definitions, and theorems used in this monograph, which are, however, rather standard and thus presented here mostly without proof, although some specific generalizations or modifications are accompanied by proofs. There are many textbooks and monographs on this subject, such as, e.g., [52, 84, 173, 175, 314, 344, 624]. A.1 Ordering A binary relation, denoted by Ä,onasetX is called an ordering if it is reflexive (i.e., x Ä x for all x 2 X), transitive (i.e., x1 Ä x2 & x2 Ä x3 imply x1 Ä x3 for all x1; x2; x3 2 X) and antisymmetric (i.e., x1 Ä x2 & x2 Ä x1 imply x1 D x2). The ordering Ä is called linear if x1 Ä x2 or x2 Ä x1 always holds for every x1; x2 2 X. An ordered set X is called directed if for every x1; x2 2 X, there is x3 2 X such that both x1 Ä x3 and x2 Ä x3. Instead of x1 Ä x2, we also write x2 x1.Byx1 < x2 we understand that x1 Ä x2 but x1 6D x2. Having two ordered sets X1 and X2 and a mapping f W X1 ! X2, we say that f is nondecreasing (respectively nonincreasing) if x1 Ä x2 implies f .x1/ Ä f .x2/ (respectively f .x1/ f .x2/). We say that x1 2 X is the greatest element of the ordered set X if x2 Ä x1 for every x2 2 X. Similarly, x1 2 X is the least element of X if x1 Ä x2 for every x2 2 X. We say that x1 2 X is maximal in the ordered set X if there is no x2 2 X such that x1 < x2. Note that the greatest element, if it exists, is always maximal but not conversely. Similarly, x1 2 X is minimal in X if there is no x2 2 X such that x1 > x2. The ordering Ä on X induces the ordering on a subset A of X, given just by the restriction of the relation Ä. We say that x1 2 X is an upper bound of A X if x2 Ä x1 for every x2 2 A. Analogously, x1 2 X is called a lower bound of A if x1 Ä x2 for every x2 2 A. If every two elements x1; x2 2 X possess both a least upper bound and a greatest lower bound, denoted respectively by sup.x1; x2/ and © Springer Science+Business Media New York 2015 579 A. Mielke, T. Roubícek,ˇ Rate-Independent Systems: Theory and Application, Applied Mathematical Sciences 193, DOI 10.1007/978-1-4939-2706-7 580 A Elements of topology and functional analysis inf.x1; x2/ and called the supremum and the infimum of fx1; x2g, then the ordered set .X; Ä/ is called a lattice. Then the supremum and the infimum exist for every finite subset and are determined uniquely because the ordering is antisymmetric. The so-called Kuratowski–Zorn lemma [343, 631] says that if every linearly ordered subset of X has an upper bound in X, then X has at least one maximal 1 element. Having a directed set and another set X, we say that .x /2 is a net in . / X if there is a mapping ! X W 7! x . Having another net xQ Q Q 2Q in X,we say that this net is finer than the net .x /2 if there is a mapping j W Q ! such Q Q that for every 2 ,wehavexQ Q D xj./Q , and moreover, for every 2 , there is Q 2 Q large enough that j.Q 1/ whenever Q 1 Q . The set of all natural numbers N ordered by the standard ordering Ä is a directed set. The nets having N (directed by this standard ordering) as the index set are called sequences. Every subsequence of a given sequence can be simultaneously understood as a finer net.2 A.2 Topology A topology T of a set X is a collection of subsets of X such that T contains the empty set and X itself, and with every finite collection of sets also their intersection, and also with every arbitrary collection of sets also their union. The elements of T are called open sets (or T-open, if we want to indicate explicitly the topology in question), while their complements are called closed.AsetX endowed with a topology T is called a topological space; sometimes we denote it by .X; T/ to refer to T explicitly. A collection T0 of subsets of X is called a base of a topology T if every T-open set is a union of elements of T0. Then T WD f [˛A˛ j8˛ W A˛ 2 T0 g is a topology (induced by the base T0). Having a subset A X, TjA WD f A \ B j B 2 T g is a topology on A, called a relative topology.Havingx 2 N X, we say that N is a neighborhood of x if there is an open set A such that x 2 A N;thesetofall neighborhoods of a point x is denoted by N .x/ or, more specifically, NT.x/ if the topology T needs to be specified. A topology T is called Hausdorff if 8x1; x2 2 X, x1 ¤ x2 9A1 2 N .x1/; A2 2 N .x2/: A1 \ A2 D;. We define the interior,theclosure, and the boundary of a set A respectively by n ˇ o ˇ int.A/ WD x 2 X ˇ 9N 2 N .x/ W N A ; (A.2.1a) 1This assertion is unfortunately highly nonconstructive unless X N, and is equivalent to the axiom ofS choice: for every set X and every collection fAxgx2X, ;¤Ax X, there is a mapping . / f W X ! x2X Ax such that f x 2 Ax for every x 2 X. 2 Indeed, having a sequence .xk/k2N and its subsequence .xk/k2N with some N N, one can put WD .N; Ä/, Q WD .N; Ä/,andj W Q ! the inclusion N N. A Elements of topology and functional analysis 581 n ˇ o ˇ cl.A/ WD x 2 X ˇ 8N 2 N .x/ W N \ A 6D; (A.2.1b) @A WD cl.A/ n int.A/: (A.2.1c) Having A B X, we say that A is dense in B if cl.A/ B. A topological space is called separable if it contains a countable subset that is dense in it. Having a net .x /2 in the topological space X, we say that it converges to a point x 2 X if for every neighborhood N of x, there is 0 2 large enough that x 2 N whenever 0; then we say also that x is the limit point of the net in question, and write lim2 x WD x or simply x ! x. This concept of convergence is called the Moore–Smith convergence [433]. Note that x 2 cl.A/ if and only if there is a net in A converging to x; in this case, we also say that x is attainable by a net from A. A point x 2 X is called a cluster point of the net .x /2 if for every neighborhood N of x and for every 0 2 , there is 0 such that x 2 N. Clearly, every limit point is a cluster point as well, but not conversely. Nevertheless, for every 3 . / cluster point x of a net x 2 , there exists a finer net fxQ Q gQ 2Q converging to x. Ordering of all topologies on a given set X is naturally by inclusion: having two topologies T1 and T2 onasetX, we say that T1 is finer than T2 or T2 is coarser than T1 if T1 T2 (or equivalently, if the identity on X is .T1; T2/-continuous). The adjectives “stronger” and “weaker” are sometimes used in place of “finer” and “coarser,” respectively. A function d W XX ! R1 is called a quasidistance on X if for all x1; x2; x3 2 X, d.x1; x2/ 0, d.x1; x2/ D 0 is equivalent to x1 D x2, and d.x1; x2/ Ä d.x1; x3/ C d.x3; x1/. A quasidistance that does not take values 1 is called a distance.Every distance d induces a topology T by a base ffx 2 X j d.x; x1/<"gjx1 2 X;">0g. A distance d W XX ! R is called a metric on X if d.x1; x2/ D d.x2; x1/ for all x1; x2 2 X. Conversely, a topology is called metrizable if there exists a metric that induces it. However, it should be emphasized that there exist nonmetrizable topologies. We say that a topological space is compact if every open cover admits a finite subcover. In terms of nets, the equivalent definition reads that every net has a cluster point. There are various useful modifications of this notion. We say that a topology is sequentially compact if every sequence in X admits a subsequence that converges in X. A metrizable topology is compact if and only if it is sequentially compact. A set A is called relatively (sequentially) compact if the closure of A is (sequentially) compact in X. A topological space is called locally (sequentially) compact if every point of it possesses a (sequentially) compact neighborhood. 3It suffices to put Q WD N .x/ directed by the ordering Äand to take, for every Q D .; / Q Q N 2 ,somexQ Q WD x 2 N with .
Load more

Recommended publications
  • 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]
  • 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).
    [Show full text]
  • Functional Analysis Lecture Notes Chapter 2. Operators on Hilbert Spaces
    FUNCTIONAL ANALYSIS LECTURE NOTES CHAPTER 2. OPERATORS ON HILBERT SPACES CHRISTOPHER HEIL 1. Elementary Properties and Examples First recall the basic definitions regarding operators. Definition 1.1 (Continuous and Bounded Operators). Let X, Y be normed linear spaces, and let L: X Y be a linear operator. ! (a) L is continuous at a point f X if f f in X implies Lf Lf in Y . 2 n ! n ! (b) L is continuous if it is continuous at every point, i.e., if fn f in X implies Lfn Lf in Y for every f. ! ! (c) L is bounded if there exists a finite K 0 such that ≥ f X; Lf K f : 8 2 k k ≤ k k Note that Lf is the norm of Lf in Y , while f is the norm of f in X. k k k k (d) The operator norm of L is L = sup Lf : k k kfk=1 k k (e) We let (X; Y ) denote the set of all bounded linear operators mapping X into Y , i.e., B (X; Y ) = L: X Y : L is bounded and linear : B f ! g If X = Y = X then we write (X) = (X; X). B B (f) If Y = F then we say that L is a functional. The set of all bounded linear functionals on X is the dual space of X, and is denoted X0 = (X; F) = L: X F : L is bounded and linear : B f ! g We saw in Chapter 1 that, for a linear operator, boundedness and continuity are equivalent.
    [Show full text]
  • Arxiv:0906.4883V2 [Math.CA] 4 Jan 2010 Ro O Olna Yeblcsses[4.W Hwblwta El’ T Helly’S That Theorem
    THE KOLMOGOROV–RIESZ COMPACTNESS THEOREM HARALD HANCHE-OLSEN AND HELGE HOLDEN Abstract. We show that the Arzelà–Ascoli theorem and Kolmogorov com- pactness theorem both are consequences of a simple lemma on compactness in metric spaces. Their relation to Helly’s theorem is discussed. The paper contains a detailed discussion on the historical background of the Kolmogorov compactness theorem. 1. Introduction Compactness results in the spaces Lp(Rd) (1 p < ) are often vital in exis- tence proofs for nonlinear partial differential equations.≤ ∞ A necessary and sufficient condition for a subset of Lp(Rd) to be compact is given in what is often called the Kolmogorov compactness theorem, or Fréchet–Kolmogorov compactness theorem. Proofs of this theorem are frequently based on the Arzelà–Ascoli theorem. We here show how one can deduce both the Kolmogorov compactness theorem and the Arzelà–Ascoli theorem from one common lemma on compactness in metric spaces, which again is based on the fact that a metric space is compact if and only if it is complete and totally bounded. Furthermore, we trace out the historical roots of Kolmogorov’s compactness theorem, which originated in Kolmogorov’s classical paper [18] from 1931. However, there were several other approaches to the issue of describing compact subsets of Lp(Rd) prior to and after Kolmogorov, and several of these are described in Section 4. Furthermore, extensions to other spaces, say Lp(Rd) (0 p < 1), Orlicz spaces, or compact groups, are described. Helly’s theorem is often≤ used as a replacement for Kolmogorov’s compactness theorem, in particular in the context of nonlinear hyperbolic conservation laws, in spite of being more specialized (e.g., in the sense that its classical version requires one spatial dimension).
    [Show full text]
  • On the Origin and Early History of Functional Analysis
    U.U.D.M. Project Report 2008:1 On the origin and early history of functional analysis Jens Lindström Examensarbete i matematik, 30 hp Handledare och examinator: Sten Kaijser Januari 2008 Department of Mathematics Uppsala University Abstract In this report we will study the origins and history of functional analysis up until 1918. We begin by studying ordinary and partial differential equations in the 18th and 19th century to see why there was a need to develop the concepts of functions and limits. We will see how a general theory of infinite systems of equations and determinants by Helge von Koch were used in Ivar Fredholm’s 1900 paper on the integral equation b Z ϕ(s) = f(s) + λ K(s, t)f(t)dt (1) a which resulted in a vast study of integral equations. One of the most enthusiastic followers of Fredholm and integral equation theory was David Hilbert, and we will see how he further developed the theory of integral equations and spectral theory. The concept introduced by Fredholm to study sets of transformations, or operators, made Maurice Fr´echet realize that the focus should be shifted from particular objects to sets of objects and the algebraic properties of these sets. This led him to introduce abstract spaces and we will see how he introduced the axioms that defines them. Finally, we will investigate how the Lebesgue theory of integration were used by Frigyes Riesz who was able to connect all theory of Fredholm, Fr´echet and Lebesgue to form a general theory, and a new discipline of mathematics, now known as functional analysis.
    [Show full text]
  • A Appendix: Linear Algebra and Functional Analysis
    A Appendix: Linear Algebra and Functional Analysis In this appendix, we have collected some results of functional analysis and matrix algebra. Since the computational aspects are in the foreground in this book, we give separate proofs for the linear algebraic and functional analyses, although finite-dimensional spaces are special cases of Hilbert spaces. A.1 Linear Algebra A general reference concerning the results in this section is [47]. The following theorem gives the singular value decomposition of an arbitrary real matrix. Theorem A.1. Every matrix A ∈ Rm×n allows a decomposition A = UΛV T, where U ∈ Rm×m and V ∈ Rn×n are orthogonal matrices and Λ ∈ Rm×n is diagonal with nonnegative diagonal elements λj such that λ1 ≥ λ2 ≥ ··· ≥ λmin(m,n) ≥ 0. Proof: Before going to the proof, we recall that a diagonal matrix is of the form ⎡ ⎤ λ1 0 ... 00... 0 ⎢ ⎥ ⎢ . ⎥ ⎢ 0 λ2 . ⎥ Λ = = [diag(λ ,...,λm), 0] , ⎢ . ⎥ 1 ⎣ . .. ⎦ 0 ... λm 0 ... 0 if m ≤ n, and 0 denotes a zero matrix of size m×(n−m). Similarly, if m>n, Λ is of the form 312 A Appendix: Linear Algebra and Functional Analysis ⎡ ⎤ λ1 0 ... 0 ⎢ ⎥ ⎢ . ⎥ ⎢ 0 λ2 . ⎥ ⎢ ⎥ ⎢ . .. ⎥ ⎢ . ⎥ diag(λ ,...,λn) Λ = ⎢ ⎥ = 1 , ⎢ λn ⎥ ⎢ ⎥ 0 ⎢ 0 ... 0 ⎥ ⎢ . ⎥ ⎣ . ⎦ 0 ... 0 where 0 is a zero matrix of the size (m − n) × n. Briefly, we write Λ = diag(λ1,...,λmin(m,n)). n Let A = λ1, and we assume that λ1 =0.Let x ∈ R be a unit vector m with Ax = A ,andy =(1/λ1)Ax ∈ R , i.e., y is also a unit vector. We n m pick vectors v2,...,vn ∈ R and u2,...,um ∈ R such that {x, v2,...,vn} n is an orthonormal basis in R and {y,u2,...,um} is an orthonormal basis in Rm, respectively.
    [Show full text]
  • AMATH 731: Applied Functional Analysis Lecture Notes
    AMATH 731: Applied Functional Analysis Lecture Notes Sumeet Khatri November 24, 2014 Table of Contents List of Tables ................................................... v List of Theorems ................................................ ix List of Definitions ................................................ xii Preface ....................................................... xiii 1 Review of Real Analysis .......................................... 1 1.1 Convergence and Cauchy Sequences...............................1 1.2 Convergence of Sequences and Cauchy Sequences.......................1 2 Measure Theory ............................................... 2 2.1 The Concept of Measurability...................................3 2.1.1 Simple Functions...................................... 10 2.2 Elementary Properties of Measures................................ 11 2.2.1 Arithmetic in [0, ] .................................... 12 1 2.3 Integration of Positive Functions.................................. 13 2.4 Integration of Complex Functions................................. 14 2.5 Sets of Measure Zero......................................... 14 2.6 Positive Borel Measures....................................... 14 2.6.1 Vector Spaces and Topological Preliminaries...................... 14 2.6.2 The Riesz Representation Theorem........................... 14 2.6.3 Regularity Properties of Borel Measures........................ 14 2.6.4 Lesbesgue Measure..................................... 14 2.6.5 Continuity Properties of Measurable Functions...................
    [Show full text]
  • Functional Analysis
    Functional Analysis Individualized Behavior Intervention for Early Education Behavior Assessments • Purpose – Analyze and understand environmental factors contributing to challenging or maladaptive behaviors – Determine function of behaviors – Develop best interventions based on function – Determine best replacement behaviors to teach student Types of Behavior Assessments • Indirect assessment – Interviews and questionnaires • Direct observation/Descriptive assessment – Observe behaviors and collect data on antecedents and consequences • Functional analysis/Testing conditions – Experimental manipulations to determine function What’s the Big Deal About Function? • Function of behavior is more important than what the behavior looks like • Behaviors can serve multiple functions Why bother with testing? • Current understanding of function is not correct • Therefore, current interventions are not working Why bother with testing? • What the heck is the function?! • Observations alone have not been able to determine function Functional Analysis • Experimental manipulations and testing for function of behavior • Conditions – Test for Attention – Test for Escape – Test for Tangible – Test for Self Stimulatory – “Play” condition, which serves as the control Test for Attention • Attention or Self Stim? • http://www.youtube.com/watch? v=dETNNYxXAOc&feature=related • Ignore student, but stay near by • Pay attention each time he screams, see if behavior increases Test for Escape • Escape or attention? • http://www.youtube.com/watch? v=wb43xEVx3W0 (second
    [Show full text]
  • Fact Sheet Functional Analysis
    Fact Sheet Functional Analysis Literature: Hackbusch, W.: Theorie und Numerik elliptischer Differentialgleichungen. Teubner, 1986. Knabner, P., Angermann, L.: Numerik partieller Differentialgleichungen. Springer, 2000. Triebel, H.: H¨ohere Analysis. Harri Deutsch, 1980. Dobrowolski, M.: Angewandte Funktionalanalysis, Springer, 2010. 1. Banach- and Hilbert spaces Let V be a real vector space. Normed space: A norm is a mapping k · k : V ! [0; 1), such that: kuk = 0 , u = 0; (definiteness) kαuk = jαj · kuk; α 2 R; u 2 V; (positive scalability) ku + vk ≤ kuk + kvk; u; v 2 V: (triangle inequality) The pairing (V; k · k) is called a normed space. Seminorm: In contrast to a norm there may be elements u 6= 0 such that kuk = 0. It still holds kuk = 0 if u = 0. Comparison of two norms: Two norms k · k1, k · k2 are called equivalent if there is a constant C such that: −1 C kuk1 ≤ kuk2 ≤ Ckuk1; u 2 V: If only one of these inequalities can be fulfilled, e.g. kuk2 ≤ Ckuk1; u 2 V; the norm k · k1 is called stronger than the norm k · k2. k · k2 is called weaker than k · k1. Topology: In every normed space a canonical topology can be defined. A subset U ⊂ V is called open if for every u 2 U there exists a " > 0 such that B"(u) = fv 2 V : ku − vk < "g ⊂ U: Convergence: A sequence vn converges to v w.r.t. the norm k · k if lim kvn − vk = 0: n!1 1 A sequence vn ⊂ V is called Cauchy sequence, if supfkvn − vmk : n; m ≥ kg ! 0 for k ! 1.
    [Show full text]
  • Arxiv:2005.07561V1 [Math-Ph]
    FABER-KRAHN INEQUALITIES FOR SCHRODINGER¨ OPERATORS WITH POINT AND WITH COULOMB INTERACTIONS VLADIMIR LOTOREICHIK AND ALESSANDRO MICHELANGELI ABSTRACT. We obtain new Faber-Krahn-type inequalities for certain perturba- tions of the Dirichlet Laplacian on a bounded domain. First, we establish a two- and three-dimensional Faber-Krahn inequality for the Schr¨odinger operator with point interaction: the optimiser is the ball with the point interaction supported at its centre. Next, we establish three-dimensional Faber-Krahn inequalities for one- and two-body Schr¨odinger operator with attractive Coulomb interactions, the optimiser being given in terms of Coulomb attraction at the centre of the ball. The proofs of such results are based on symmetric decreasing rearrangement and Steiner rearrangement techniques; in the first model a careful analysis of certain monotonicity properties of the lowest eigenvalue is also needed. 1. Background and outline In this work we produce two types of generalisations of a famous, one century old, optimisation result due to G. Faber [31] and E. Krahn [49]. Whereas our applications concern two distinct operators of interest, the conceptual scheme of the proofs and the technical tools utilised are similar in both cases, which is the reason we should like to present both results on the same footing. In its original formulation Faber-Krahn inequality states that amongst all domains Ω ⊂ Rd, d ≥ 2, with the same given finite volume, the lowest (principal) eigen- value of the negative Dirichlet Laplacian is minimised by the ball. This is an archetypal result in the vast and ever growing field of variational methods for eigenvalue approximation and spectral optimisation.
    [Show full text]
  • A Spectral Theory of Linear Operators on Rigged Hilbert Spaces
    A spectral theory of linear operators on rigged Hilbert spaces under analyticity conditions Institute of Mathematics for Industry, Kyushu University, Fukuoka, 819-0395, Japan Hayato CHIBA 1 Jul 29, 2011; last modified Sep 12, 2014 Abstract A spectral theory of linear operators on rigged Hilbert spaces (Gelfand triplets) is devel- oped under the assumptions that a linear operator T on a Hilbert space is a perturbation of a selfadjoint operator, and the spectral measure of the selfadjoint operatorH has an an- alytic continuation near the real axis in some sense. It is shown that there exists a dense 1 subspace X of such that the resolvent (λ T)− φ of the operator T has an analytic H − continuation from the lower half plane to the upper half plane as an X′-valued holomor- phic function for any φ X, even when T has a continuous spectrum on R, where X′ is ∈ a dual space of X. The rigged Hilbert space consists of three spaces X X′. A ⊂ H ⊂ generalized eigenvalue and a generalized eigenfunction in X′ are defined by using the an- alytic continuation of the resolvent as an operator from X into X′. Other basic tools of the usual spectral theory, such as a spectrum, resolvent, Riesz projection and semigroup are also studied in terms of a rigged Hilbert space. They prove to have the same properties as those of the usual spectral theory. The results are applied to estimate asymptotic behavior of solutions of evolution equations. Keywords: generalized eigenvalue; resonance pole; rigged Hilbert space; Gelfand triplet; generalized function 1 Introduction A spectral theory of linear operators on topological vector spaces is one of the central issues in functional analysis.
    [Show full text]
  • A Practical Guide to Compact Infinite Dimensional Parameter Spaces
    A Practical Guide to Compact Infinite Dimensional Parameter Spaces∗ Joachim Freybergery Matthew A. Mastenz May 24, 2018 Abstract Compactness is a widely used assumption in econometrics. In this paper, we gather and re- view general compactness results for many commonly used parameter spaces in nonparametric estimation, and we provide several new results. We consider three kinds of functions: (1) func- tions with bounded domains which satisfy standard norm bounds, (2) functions with bounded domains which do not satisfy standard norm bounds, and (3) functions with unbounded do- mains. In all three cases we provide two kinds of results, compact embedding and closedness, which together allow one to show that parameter spaces defined by a k · ks-norm bound are compact under a norm k · kc. We illustrate how the choice of norms affects the parameter space, the strength of the conclusions, as well as other regularity conditions in two common settings: nonparametric mean regression and nonparametric instrumental variables estimation. JEL classification: C14, C26, C51 Keywords: Nonparametric Estimation, Sieve Estimation, Trimming, Nonparametric Instrumental Variables ∗This paper was presented at Duke, the 2015 Triangle Econometrics Conference, and the 2016 North American and China Summer Meetings of the Econometric Society. We thank audiences at those seminars as well as Bruce Hansen, Kengo Kato, Jack Porter, Yoshi Rai, and Andres Santos for helpful conversations and comments. We also thank the editor and the referees for thoughtful feedback which improved this paper. yDepartment of Economics, University of Wisconsin-Madison, William H. Sewell Social Science Building, 1180 Observatory Drive, Madison, WI 53706, USA. [email protected] zCorresponding author.
    [Show full text]