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”.