DOCSLIB.ORG

On the Origin and Early History of Functional Analysis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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. Acknowledgements First of all, I would like to give my sincerest gratitudes to Sten Kaijser, not only for supervising this thesis, but also for being my menthor during my years at the university. If it were not for him, I would have followed my original plan and study theoretical philosophy instead of mathematics. For preventing this, I am grateful. Secondly, I am grateful for the help of Gunnar Berg who provided me with helpful comments and criticism to improve this thesis. Finally I give my gratitudes to Olivier for interesting conversations and help with French translations and bad grammar. Contents 1 Introduction 4 2 Differential equations 5 2.1 Linear ordinary differential equations . 5 2.2 Partial differential equations . 5 2.3 Spectral theory . 6 3 Integral equations 10 3.1 Origins in applications . 10 3.2 Potential theory and electrostatics . 11 3.3 The connection between Differential and Integral equations . 13 4 Passing to the infinite 16 4.1 Pre-linear algebra . 16 4.2 Infinite systems and determinants . 16 4.3 General theory . 20 5 The concept of space in the 19th century 23 6 Fredholm on Integral equations 25 7 Hilbert on Spectral theory 30 8 Finalizing the concept of space 39 8.1 A new way of mathematics – Abstraction prior to problem solving . 39 8.2 Adding structure to abstract spaces – The introduction of Topology . 39 9 Fr´echet on metric spaces 41 9.1 Synthetic geometry – Euclidean geometry in function spaces . 43 10 Lebesgue on Integration theory 45 11 The creation of modern Functional Analysis 48 11.1 Spectral theory of compact operators . 52 A Solution of the Dirichlet and Neumann problems by Fredholm’s metod 57 A.1 The interior Dirichlet problem . 57 A.2 The exterior Dirichlet problem . 58 A.3 The interior Neumann problem . 59 A.4 The exterior Neumann problem . 60 3 1 Introduction Functional analysis is the branch of mathematics where vector spaces and operators on them are in focus. In linear algebra, the discussion is about finite dimensional vector spaces over any field of scalars. The functions are linear mappings which can be viewed as matrices with scalar entries. If the functions are mappings from a vector space to itself, the functions are called operators and they are represented by square matrices. In functional analysis, the vector spaces are in general infinite dimensional and not all operators on them can be represented by matrices. Hence the theory becomes more complicated, but nonetheless there are many similarities. Functional analysis has its origin in ordinary and partial differential equations, and in the beginning of the 20th century it started to form a discipline of its own via integral equations. However, for a long time there were doubts wether the mathematical theory was rich enough. Despite the efforts of many prominent mathematicians, it was not sure if there were sufficiently many functionals to support a good theory, and it was not until 1920 that the question was finally settled with the celebrated Hahn–Banach theorem. Seen from the modern point of view, functional analysis can be considered as a gener- alization of linear algebra. However, from a historical point of view, the theory of linear algebra was not developed enough to provide a basis for functional analysis at its time of creation. Thus, to study the history of functional analysis we need to investigate which concepts of mathematics that needed to be completed in order to get a theory rigorous enough to support it. Those concepts turn out to be functions, limits and set theory. For a long time, the definition of a function was due to Euler in his Introductio in Analysin Infinitorum from 1748 which read: ”A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities.”. For a detailed discussion about the problems concerning this definition, see [11]. For the purpose of this report, it is enough to say that the entire focus of this definition is on the function itself, and the properties of this particular function. What lead to the success of functional analysis was that the focus was lifted from the function, and shifted to the algebraic properties of sets of functions – The algebraization of analysis. The process of algebraization led mathematicians to study sets of functions where the functions are nothing more than abstract points in the set. At the same time as the theory is very concrete and applicable to physical problems, it can be presented in a very abstract way. Some proofs and results are significantly simplified by introducing the axiom of choice, Zorn’s lemma or the Baire category theorem – some of the most abstract concepts in set theory. The main theorems are 1. The Hahn-Banach theorem by Hans Hahn and Stephan Banach, which states that there are sufficiently many continuous functionals on every normed space to make the theory of dual spaces and adjoint operators interesting 2. The uniform boundedness principle or the Banach-Steinhaus theorem by Banach and Hugo Steinhaus, which states that for any family of continuous linear operators on a Banach space, pointwise boundedness is equivalent to boundedness in the operator norm 3. The open mapping theorem or the Banach-Schauder theorem by Banach and Juliusz Pawel Schauder, which classifies the open mappings between two Banach spaces 4 2 Differential equations 2.1 Linear ordinary differential equations Due to the unclear notion of a function during the end of the 18th and the beginning of the 19th century, one thought of a function in the same way as we today would think of an analytic function. That is, it was asumed that around each point x0, the function was equal to a power series in x − x0. Taking derivatives of this function was equal to taking derivatives of the terms in the series expansion. In general convergence was not considered. The common recipe for solving a differential equation y(n) = F (x, y, y0, y00, . , y(n−1)) (1) P∞ k would then be to substitute the power series y = 0 ck(x − x0) and its termwise deriva- tives, into (1). Identifying the series on both sides would then decide ck, for k ≥ n, as a function of c0, c1, . , ck−1. The usefulness of this metod was restricted to rather simple differential equations such as the linear equation y0 = a(x)y + b(x) for which the solution had been known since the 17th century. It was not until after 1760 that a general study of ordinary linear differential equations of arbitrary order began. [5] 2.2 Partial differential equations In the 18th century the development was triggered by physical problems, and one of the best examples of this fact is the theory of partial differential equations. In 1747, Jean le Rond d’Alembert (1717 – 1783) published a paper which proposed a solution to the vibrating string problem. Since the position of any point on the string is depending on both time and position, a function describing the shape of the string must depend on two variables, y = f(x, t). d’Alembert considered the string to be composed of infinitely many small parts, each with infinitely small mass, and used Newton’s laws of motion to derive a partial differential equation for the shape of the string, now called the wave equation, ∂2y ∂2y = c2 , (2) ∂t2 ∂x2 where c is a known function of x and constant if the mass of the string is homogeneous. d’Alembert considered the special case when c2 = 1 and by the change of variables X = x − t, Y = x + t he reduced (2) to ∂2y = 0. (3) ∂X∂Y From (3) d’Alembert concluded that the solution of (2) was y(x, t) = f(x − t) + g(x + t) where f and g are arbitrary twice differentiable functions. [11] This caused quite a controversy because of the deception that a function had to be something very concrete, and not ”arbitrary”.
Load more

Recommended publications
  • 1 Bounded and Unbounded Operators
    1 1 Bounded and unbounded operators 1. Let X, Y be Banach spaces and D 2 X a linear space, not necessarily closed. 2. A linear operator is any linear map T : D ! Y . 3. D is the domain of T , sometimes written Dom (T ), or D (T ). 4. T (D) is the Range of T , Ran(T ). 5. The graph of T is Γ(T ) = f(x; T x)jx 2 D (T )g 6. The kernel of T is Ker(T ) = fx 2 D (T ): T x = 0g 1.1 Operations 1. aT1 + bT2 is defined on D (T1) \D (T2). 2. if T1 : D (T1) ⊂ X ! Y and T2 : D (T2) ⊂ Y ! Z then T2T1 : fx 2 D (T1): T1(x) 2 D (T2). In particular, inductively, D (T n) = fx 2 D (T n−1): T (x) 2 D (T )g. The domain may become trivial. 3. Inverse. The inverse is defined if Ker(T ) = f0g. This implies T is bijective. Then T −1 : Ran(T ) !D (T ) is defined as the usual function inverse, and is clearly linear. @ is not invertible on C1[0; 1]: Ker@ = C. 4. Closable operators. It is natural to extend functions by continuity, when possible. If xn ! x and T xn ! y we want to see whether we can define T x = y. Clearly, we must have xn ! 0 and T xn ! y ) y = 0; (1) since T (0) = 0 = y. Conversely, (1) implies the extension T x := y when- ever xn ! x and T xn ! y is consistent and defines a linear operator.
    [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]
  • Algebraic Generality Vs Arithmetic Generality in the Controversy Between C
    Algebraic generality vs arithmetic generality in the controversy between C. Jordan and L. Kronecker (1874). Frédéric Brechenmacher (1). Introduction. Throughout the whole year of 1874, Camille Jordan and Leopold Kronecker were quarrelling over two theorems. On the one hand, Jordan had stated in his 1870 Traité des substitutions et des équations algébriques a canonical form theorem for substitutions of linear groups; on the other hand, Karl Weierstrass had introduced in 1868 the elementary divisors of non singular pairs of bilinear forms (P,Q) in stating a complete set of polynomial invariants computed from the determinant |P+sQ| as a key theorem of the theory of bilinear and quadratic forms. Although they would be considered equivalent as regard to modern mathematics (2), not only had these two theorems been stated independently and for different purposes, they had also been lying within the distinct frameworks of two theories until some connections came to light in 1872-1873, breeding the 1874 quarrel and hence revealing an opposition over two practices relating to distinctive cultural features. As we will be focusing in this paper on how the 1874 controversy about Jordan’s algebraic practice of canonical reduction and Kronecker’s arithmetic practice of invariant computation sheds some light on two conflicting perspectives on polynomial generality we shall appeal to former publications which have already been dealing with some of the cultural issues highlighted by this controversy such as tacit knowledge, local ways of thinking, internal philosophies and disciplinary ideals peculiar to individuals or communities [Brechenmacher 200?a] as well as the different perceptions expressed by the two opponents of a long term history involving authors such as Joseph-Louis Lagrange, Augustin Cauchy and Charles Hermite [Brechenmacher 200?b].
    [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]
  • Continuous Nowhere Differentiable Functions
    2003:320 CIV MASTER’S THESIS Continuous Nowhere Differentiable Functions JOHAN THIM MASTER OF SCIENCE PROGRAMME Department of Mathematics 2003:320 CIV • ISSN: 1402 - 1617 • ISRN: LTU - EX - - 03/320 - - SE Continuous Nowhere Differentiable Functions Johan Thim December 2003 Master Thesis Supervisor: Lech Maligranda Department of Mathematics Abstract In the early nineteenth century, most mathematicians believed that a contin- uous function has derivative at a significant set of points. A. M. Amp`ereeven tried to give a theoretical justification for this (within the limitations of the definitions of his time) in his paper from 1806. In a presentation before the Berlin Academy on July 18, 1872 Karl Weierstrass shocked the mathematical community by proving this conjecture to be false. He presented a function which was continuous everywhere but differentiable nowhere. The function in question was defined by ∞ X W (x) = ak cos(bkπx), k=0 where a is a real number with 0 < a < 1, b is an odd integer and ab > 1+3π/2. This example was first published by du Bois-Reymond in 1875. Weierstrass also mentioned Riemann, who apparently had used a similar construction (which was unpublished) in his own lectures as early as 1861. However, neither Weierstrass’ nor Riemann’s function was the first such construction. The earliest known example is due to Czech mathematician Bernard Bolzano, who in the years around 1830 (published in 1922 after being discovered a few years earlier) exhibited a continuous function which was nowhere differen- tiable. Around 1860, the Swiss mathematician Charles Cell´erieralso discov- ered (independently) an example which unfortunately wasn’t published until 1890 (posthumously).
    [Show full text]
  • The History of the Abel Prize and the Honorary Abel Prize the History of the Abel Prize
    The History of the Abel Prize and the Honorary Abel Prize The History of the Abel Prize Arild Stubhaug On the bicentennial of Niels Henrik Abel’s birth in 2002, the Norwegian Govern- ment decided to establish a memorial fund of NOK 200 million. The chief purpose of the fund was to lay the financial groundwork for an annual international prize of NOK 6 million to one or more mathematicians for outstanding scientific work. The prize was awarded for the first time in 2003. That is the history in brief of the Abel Prize as we know it today. Behind this government decision to commemorate and honor the country’s great mathematician, however, lies a more than hundred year old wish and a short and intense period of activity. Volumes of Abel’s collected works were published in 1839 and 1881. The first was edited by Bernt Michael Holmboe (Abel’s teacher), the second by Sophus Lie and Ludvig Sylow. Both editions were paid for with public funds and published to honor the famous scientist. The first time that there was a discussion in a broader context about honoring Niels Henrik Abel’s memory, was at the meeting of Scan- dinavian natural scientists in Norway’s capital in 1886. These meetings of natural scientists, which were held alternately in each of the Scandinavian capitals (with the exception of the very first meeting in 1839, which took place in Gothenburg, Swe- den), were the most important fora for Scandinavian natural scientists. The meeting in 1886 in Oslo (called Christiania at the time) was the 13th in the series.
    [Show full text]
  • Floer Homology on Symplectic Manifolds
    Floer Homology on Symplectic Manifolds KWONG, Kwok Kun A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Philosophy in Mathematics c The Chinese University of Hong Kong August 2008 The Chinese University of Hong Kong holds the copyright of this thesis. Any person(s) intending to use a part or whole of the materials in the thesis in a proposed publication must seek copyright release from the Dean of the Graduate School. Thesis/Assessment Committee Professor Wan Yau Heng Tom (Chair) Professor Au Kwok Keung Thomas (Thesis Supervisor) Professor Tam Luen Fai (Committee Member) Professor Dusa McDuff (External Examiner) Floer Homology on Symplectic Manifolds i Abstract The Floer homology was invented by A. Floer to solve the famous Arnold conjecture, which gives the lower bound of the fixed points of a Hamiltonian symplectomorphism. Floer’s theory can be regarded as an infinite dimensional version of Morse theory. The aim of this dissertation is to give an exposition on Floer homology on symplectic manifolds. We will investigate the similarities and differences between the classical Morse theory and Floer’s theory. We will also explain the relation between the Floer homology and the topology of the underlying manifold. Floer Homology on Symplectic Manifolds ii ``` ½(ÝArnold øî×bÛñøÝFÝóê ×Íì§A. FloerxñÝFloer!§¡XÝ9 Floer!§¡Ú Morse§¡Ý×ÍP§îÌÍÍ¡Zº EøîÝFloer!®×Í+&ƺD¡BÎMorse§¡ Floer§¡Ý8«õ! ¬ÙÕFloer!ÍXòøc RÝn; Floer Homology on Symplectic Manifolds iii Acknowledgements I would like to thank my advisor Prof. Thomas Au Kwok Keung for his encouragement in writing this thesis. I am grateful to all my teachers.
    [Show full text]
  • Basic Theory of Fredholm Operators Annali Della Scuola Normale Superiore Di Pisa, Classe Di Scienze 3E Série, Tome 21, No 2 (1967), P
    ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze MARTIN SCHECHTER Basic theory of Fredholm operators Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3e série, tome 21, no 2 (1967), p. 261-280 <http://www.numdam.org/item?id=ASNSP_1967_3_21_2_261_0> © Scuola Normale Superiore, Pisa, 1967, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction 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/ BASIC THEORY OF FREDHOLM OPERATORS (*) MARTIN SOHECHTER 1. Introduction. " A linear operator A from a Banach space X to a Banach space Y is called a Fredholm operator if 1. A is closed 2. the domain D (A) of A is dense in X 3. a (A), the dimension of the null space N (A) of A, is finite 4. .R (A), the range of A, is closed in Y 5. ~ (A), the codimension of R (A) in Y, is finite. The terminology stems from the classical Fredholm theory of integral equations. Special types of Fredholm operators were considered by many authors since that time, but systematic treatments were not given until the work of Atkinson [1]~ Gohberg [2, 3, 4] and Yood [5]. These papers conside- red bounded operators.
    [Show full text]
  • History of Nonlinear Oscillations Theory in France (1880–1940)
    Archimedes 49 New Studies in the History and Philosophy of Science and Technology Jean-Marc Ginoux History of Nonlinear Oscillations Theory in France (1880–1940) History of Nonlinear Oscillations Theory in France (1880–1940) Archimedes NEW STUDIES IN THE HISTORY AND PHILOSOPHY OF SCIENCE AND TECHNOLOGY VOLUME 49 EDITOR JED Z. BUCHWALD, Dreyfuss Professor of History, California Institute of Technology, Pasadena, USA. ASSOCIATE EDITORS FOR MATHEMATICS AND PHYSICAL SCIENCES JEREMY GRAY, The Faculty of Mathematics and Computing, The Open University, UK. TILMAN SAUER, Johannes Gutenberg University Mainz, Germany ASSOCIATE EDITORS FOR BIOLOGICAL SCIENCES SHARON KINGSLAND, Department of History of Science and Technology, Johns Hopkins University, Baltimore, USA. MANFRED LAUBICHLER, Arizona State University, USA ADVISORY BOARD FOR MATHEMATICS, PHYSICAL SCIENCES AND TECHNOLOGY HENK BOS, University of Utrecht, The Netherlands MORDECHAI FEINGOLD, California Institute of Technology, USA ALLAN D. FRANKLIN, University of Colorado at Boulder, USA KOSTAS GAVROGLU, National Technical University of Athens, Greece PAUL HOYNINGEN-HUENE, Leibniz University in Hannover, Germany TREVOR LEVERE, University of Toronto, Canada JESPER LÜTZEN, Copenhagen University, Denmark WILLIAM NEWMAN, Indiana University, Bloomington, USA LAWRENCE PRINCIPE, The Johns Hopkins University, USA JÜRGEN RENN, Max Planck Institute for the History of Science, Germany ALEX ROLAND, Duke University, USA ALAN SHAPIRO, University of Minnesota, USA NOEL SWERDLOW, California Institute of Technology,
    [Show full text]
  • Dissipative Operators in a Banach Space
    Pacific Journal of Mathematics DISSIPATIVE OPERATORS IN A BANACH SPACE GUNTER LUMER AND R. S. PHILLIPS Vol. 11, No. 2 December 1961 DISSIPATIVE OPERATORS IN A BANACH SPACE G. LUMER AND R. S. PHILLIPS 1. Introduction* The Hilbert space theory of dissipative operators1 was motivated by the Cauchy problem for systems of hyperbolic partial differential equations (see [5]), where a consideration of the energy of, say, an electromagnetic field leads to an L2 measure as the natural norm for the wave equation. However there are many interesting initial value problems in the theory of partial differential equations whose natural setting is not a Hilbert space, but rather a Banach space. Thus for the heat equation the natural measure is the supremum of the temperature whereas in the case of the diffusion equation the natural measure is the total mass given by an Lx norm. In the present paper a suitable extension of the theory of dissipative operators to arbitrary Banach spaces is initiated. An operator A with domain ®(A) contained in a Hilbert space H is called dissipative if (1.1) re(Ax, x) ^ 0 , x e ®(A) , and maximal dissipative if it is not the proper restriction of any other dissipative operator. As shown in [5] the maximal dissipative operators with dense domains precisely define the class of generators of strongly continuous semi-groups of contraction operators (i.e. bounded operators of norm =§ 1). In the case of the wave equation this furnishes us with a description of all solutions to the Cauchy problem for which the energy is nonincreasing in time.
    [Show full text]
  • Transcendental Numbers
    INTRODUCTION TO TRANSCENDENTAL NUMBERS VO THANH HUAN Abstract. The study of transcendental numbers has developed into an enriching theory and constitutes an important part of mathematics. This report aims to give a quick overview about the theory of transcen- dental numbers and some of its recent developments. The main focus is on the proof that e is transcendental. The Hilbert's seventh problem will also be introduced. 1. Introduction Transcendental number theory is a branch of number theory that concerns about the transcendence and algebraicity of numbers. Dated back to the time of Euler or even earlier, it has developed into an enriching theory with many applications in mathematics, especially in the area of Diophantine equations. Whether there is any transcendental number is not an easy question to answer. The discovery of the first transcendental number by Liouville in 1851 sparked up an interest in the field and began a new era in the theory of transcendental number. In 1873, Charles Hermite succeeded in proving that e is transcendental. And within a decade, Lindemann established the tran- scendence of π in 1882, which led to the impossibility of the ancient Greek problem of squaring the circle. The theory has progressed significantly in recent years, with answer to the Hilbert's seventh problem and the discov- ery of a nontrivial lower bound for linear forms of logarithms of algebraic numbers. Although in 1874, the work of Georg Cantor demonstrated the ubiquity of transcendental numbers (which is quite surprising), finding one or proving existing numbers are transcendental may be extremely hard. In this report, we will focus on the proof that e is transcendental.
    [Show full text]