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History of Functional Analysis
History of Functional Analysis presents functional analysis as a rather complex blend of algebra and topology, with its evolution influenced by the development of these two branches of mathematics. The book adopts a narrower definition--one that is assumed to satisfy various algebraic and topological conditions. A moment of reflections shows that this already covers a large part of modern analysis, in particular, the theory of partial differential equations. This volume comprises nine chapters, the first of which focuses on linear differential equations and the Sturm-Liouville problem. The succeeding chapters go on to discuss the crypto-integral equations, including the Dirichlet principle and the Beer-Neumann method; the equation of vibrating membranes, including the contributions of Poincare and H.A. Schwarz's 1885 paper; and the idea of infinite dimension. Other chapters cover the crucial years and the definition of Hilbert space, including Fredholm's discovery and the contributions of Hilbert; duality and the definition of normed spaces, including the Hahn-Banach theorem and the method of the gliding hump and Baire category; spectral theory after 1900, including the theories and works of F. Riesz, Hilbert, von Neumann, Weyl, and Carleman; locally convex spaces and the theory of distributions; and applications of functional analysis to differential and partial differential equations. This book will be of interest to practitioners in the fields of mathematics and statistics.
- GenresMathematics
320 pages, Kindle Edition
First published January 1, 1981
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July 28, 2020
As a rule, historians and philosophers of mathematics lack the technical competence to enter into debates on issues at the research frontier and, in consequence, tend to circle around a small number of topics they can understand fairly well, such as the endlessly controverted question as to whether Leibniz had a rigorous understanding of infinitesimals along the lines of Robinson’s non-standard analysis, or not. But what the student of mathematics who purposes to launch his own career would really want is a guide to what it is important to know and to how we arrived at the present state of the art. One would have hoped that the historians and philosophers could avail themselves of their expertise in order to throw light on these concerns. Happily, one has no cause for such reservations with the present review of the history of functional analysis by the eminent French mathematician Jean Dieudonné.
The author was one of the luminaries of the Bourbaki clique of French mathematicians active during the mid-twentieth century, and contributed substantially towards its pedagogical goals by writing a number of textbooks in its distinctive style, including one on analysis. Thus, he is ideally suited to undertake the present historical work. Dieudonné has done his homework and knows the literature minutely, all the way back to its beginnings early in the nineteenth century. The reader will find it a rewarding exercise to follow the author’s cue and work backward from the compact, polished presentations in modern textbooks one knows to the original papers. The point is to see the context in which the important ideas arose long before they were honed into the powerful tools over which we dispose nowadays. Sometimes, it can be instructive to set these later refinements aside, just as one can best learn how to orient oneself when hiking in the wilderness by turning off the GPS and relying on an old-fashioned compass and the position of the sun in the sky. Afterwards, one will be free to avail oneself of modern methods with greater discernment. Dieudonné has meticulously reconstructed all the major developments in functional analysis up to the time of his writing in the 1970’s and tells its story well, replete with abundant citations to the primary sources.
What is in the text? The roots of what would come to be known as functional analysis lie in the use of Fourier expansions, as part of which one was led to nascent ideas of spectral theory that were to be enlarged upon by the Sturm-Liouville theory, involving ordinary differential equations of second order in a slightly more general form. The effort after 1880 to apply these ideas in the setting of partial differential equations would stimulate the modern development of spectral theory covered in chapter three. But before going there, Dieudonné swerves into a detour in chapter two on integral, or crypto-integral equations. The original motivation lies in celestial mechanics, where one would like to justify a perturbative approach leading to successive approximations of the true solution in powers of the expansion parameter, presumed to be small. When Cauchy did so in 1835, he arrived at a hierarchy of integral equations, which he showed was convergent, the first time ever that anyone had found a solution to a general type of differential equation (i.e., other than one that can be given in explicit form); see pp. 22-24. Another route leading to integral equations was the potential theory of Green’s. Dieudonné then delves into the convoluted history of the Dirichlet problem during the latter half of the nineteenth century, culminating in Hilbert’s dramatic resolution of the problem in 1899 on the basis of novel direct techniques. Chapter two closes with a presentation of the Beer-Neumann method of constructing solutions to the Dirichlet problem in terms of the boundary via recursive integral equations (later to be called a Fredholm integral equation of second kind).
Chapter three is devoted to the contributions of Schwarz, Weber and Poincaré to the vibrating membrane, the logical extension of the ordinary differential equation for the vibrating string. Although it concerns only a single case, the vibrating membrane served as an ideal testing ground in which to adumbrate the beginnings of spectral theory, which was due to take off with Hilbert’s fundamental paper of 1906. Before turning to the glory days of the first decade of the twentieth century, though, Dieudonné takes a detour in chapter four on the very idea of infinite dimension. He describes in detail the passage from finite-dimensional linear algebra to the infinite-dimensional case. At the hands of Le Roux and Volterra, a general theory of integral equations emerged as a result. It was a good place to encounter the topological issues surrounding the circumstance that there can be several distinct ways for a sequence of functions fₙ to tend to a limit f (for instance, convergence in mean square). Ascoli’s notion of equicontinuity and the concept of an operator that maps a given function into another function arose in this context, but would soon come to be characteristic of the new field of functional analysis as a whole: the relevance of an appropriate choice of topology (what is a trivial issue in finitely many dimensions) and the ubiquitous role of operators on spaces of functions.
Fredholm, Fréchet and Hilbert established the foundations of modern functional analysis; Dieudonné: ‘Between 1900 and 1910, there was a sudden crystallization of all the ideas and methods which had been slowly accumulating during the XIXth century...essentially due to the publication of four fundamental papers: Fredholm’s 1900 paper on integral equations; Lebesgue’s thesis of 1902 on integration; Hilbert’s paper and 1906 on spectral theory and Fréchet’s thesis of 1906 on metric spaces’ (p. 97). The reader will find it eye-opening to follow the path developments took, involving the introduction of such concepts as the Fredholm determinant, resolvent kernels and quadratic forms. Often enough, we can easily recognize these concepts as simple special cases of the more powerful formalism one will have acquired from a course of lectures on modern functional analysis. But this is to look at the history backwards, from our vantage point at its denouement—precisely why the study of the history of mathematics can be such an enjoyable experience! Chapter five closes with an extended reflection on the ‘confluence of geometry, topology and analysis’ thereby conjured up. For Dieudonné, this constellation of ideas is very significant as for the first time, mathematicians began to think in terms of abstract structures (Hausdorff’s axiomatic topology, Fréchet’s metric spaces etc.) and to analyze their properties at the structural level, what calls for a leap in the capacity for abstraction and what would be characteristic of the trend of mathematics all through the following century (especially for the Bourbakians, among whom the author numbers himself), as we see in evident profusion with the rise of category theory and homological algebra after the second world war.
To get a sense of the intellectual excitement involved, consider Cayley’s view of linear algebra as a theory of n-tuples; now, due to the Riesz-Fischer theorem, square-integrable functions can be identified with elements of the space of sequences of numbers (or n-tuples as n tends to infinity). Why Cayley’s concept is inadequate, in general: no such identification is possible for C([a,b]); there, one has to handle continuous functions directly and not through coordinates, as Dieudonné explains (otherwise the Hahn-Banach theorem would be a triviality). At this juncture, functional analysis opens into a wonderful world displaying qualitatively new phenomena which Riesz was among the first to explore, with his concept of the dual space of continuous linear functionals and its potential property of reflexivity. Thus, richer possibilities that defy geometrical intuition fashioned upon finite-dimensional space supervene. The theory of duality becomes subtle and non-trivial as soon as one goes to Banach spaces that are not Hilbert spaces, such as Lᵖ(μ) for p≠2. Another classic contribution is Riesz’ 1918 paper on spectral theory of operators in Banach spaces. Weyl’s theory of singular integral kernels and Carleman kernels represent deep results that probe the intricacies unleashed by going to more general function spaces, not representable in terms of sequences of numbers. Dieudonné’s treatment of these topics is, as usual, generously detailed. For instance, he reconstructs the steps issuing in Hilbert’s discovery of operators having not just pure point but also continuous spectrum (as with the quantum-mechanical model of the hydrogen atom to appear in the 1920’s, where, in addition to the bound states of negative energy, there is a whole continuum of scattering states at every positive value of the energy).
The high point of chapter seven is von Neumann’s clever theory of unbounded operators and their self-adjoint extensions (the need for an analysis of the problem of extension of the domain arises because an unbounded operator can be defined only on a dense subset). It is worth dilating on the comparison of von Neumann to Dirac for a while in order to highlight the former’s vastly greater stature as a mathematician. The latter, as is known, was one of the codiscoverers of the quantum mechanics in its meteoric arrival on the scene in 1924-1925 (along with Heisenberg, Born, Jordan and Schrödinger). Dirac’s version of the theory emphasizes the analogy with the commutative algebra of observable functions on phase space in classical mechanics (c-numbers) and the passage to the non-commutative algebra of observable self-adjoint operators in quantum mechanics (q-numbers), under which the Poisson bracket goes over into the commutator. Dirac also, more than anyone else, taught physicists to think in terms of vectors in Hilbert space (the wavefunction). If one stays at the heuristic level, the basic formalism of quantum mechanics, including scattering theory, stationary perturbation theory and time-dependent perturbation theory, is almost childishly simple. This explains how Born, Heisenberg and Jordan were able to hammer it out in a matter of months in the so-called Dreimännerarbeit [Zeitschrift für Physik 35, 557-615 (1926)]. There is, indeed, almost no real physics in Dirac’s Principles of Quantum Mechanics (for a superior exposition of heuristic quantum mechanics, both in terms of style and in terms of content, see the two volumes of Cohen-Tannoudji; the French gift for clear thought counts for something). Mathematical physicists under the tutelage of Dirac were accustomed to treat operators loosely as if they were matrices with a countable or uncountable basis (take a look at the figures in §3.3 on continuous spectra in the paper just cited). This is scarcely valid from a mathematical point of view. As Dieudonné himself tells us: ‘But if H is not maximal (i.e. both defects are > 0), any hermitian operator which extends H obviously has the same matrix (aₙₘ); and von Neumann showed in great detail how this lack of “one-to-oneness” in the correspondence between matrices and operators led to the weirdest pathology, convincing once for all the analysts that matrices were a totally inadequate tool in spectral theory’ (p. 181). Von Neumann is the one who, all but single-handedly, worked out the rigorous formulation of the quantum mechanics in terms of Hilbert spaces, published in 1932 as the Mathematische Grundlagen der Quantenmechanik.
But von Neumann did not stop there. He went on, in collaboration with Murry, to establish the foundations of the theory of operator algebras, of which Dieudonné remarks, ‘By the wealth and novelty of their techniques and their results, these wonderful papers are certainly the most profound and most difficult which von Neumann ever wrote; they revealed a large number of completely unsuspected phenomena’ (p. 183). The right way of viewing this theory is as a non-commutative generalization of Lebesgue’s measure theory. As Dieudonné observes, it must be seen as a great surprise that von Neumann and Murry could have completed their work years before the elementary concepts of the theory of normed algebras (viz., Banach and C*-algebras) had been elucidated. In the last two sections of chapter seven, Dieudonné covers these and later post-war developments, such as Gelfand’s theorem, the structural theory of Banach algebras and harmonic analysis on locally compact groups, along with the remarkable duality theory that applies to them. The famous names associated with these ideas are those of Frobenius, Weyl, Gelfand and Pontryagin.
Chapter eight concerns locally convex spaces, a generalization of the concept of metric space and about the farthest one can go and still have good theorems. Herein lies the natural setting for Laurent Schwartz’s theory of distributions, which supplies at last a rigorous sense to Dirac’s delta function and proves to be of the utmost value in investigating differential and partial differential equations (to which Dieudonné provides an introduction in chapter nine).
Closing remarks: first, keep in mind that the reader needs to be already fairly conversant with the ideas of functional analysis to profit from this book. Otherwise, one will fail to see the point of much of what Dieudonné has to say. Within space limitations, it is possible only to give precise definitions and a sketch of proof; there is little, if any, discussion of concrete examples beyond the mention that they exist and demonstrate a property or the impossibility of carrying out a certain construction in general. Yet, Dieudonné does perform the invaluable service of delineating a connected narrative of a broad swath of what is by now an extensive discipline and has conveniently assembled references to the primary literature. Compared to what one may be used to from general surveys of the history of mathematics, Dieudonné enters into microscopic detail when sketching the steps along the way. What one can get out of reading him: perspective on how the field of modern functional analysis fits together; what it is important to know and in what order; and a few helpful judgments on quality. In view whereof, warmly to be recommended to the serious and diligent student.
P.S. A minor gripe: why, after the advent of convenient computerized type-setting, could the publisher North-Holland not have invested the seemingly minor sum to have somebody recompose the typewritten manuscript from back in 1981 instead of merely photocopying it? The typewriter font lends the text an unattractive appearance (especially with displayed equations), yet it does not hinder comprehension as much as one might have feared, once one becomes engaged with the material.
The author was one of the luminaries of the Bourbaki clique of French mathematicians active during the mid-twentieth century, and contributed substantially towards its pedagogical goals by writing a number of textbooks in its distinctive style, including one on analysis. Thus, he is ideally suited to undertake the present historical work. Dieudonné has done his homework and knows the literature minutely, all the way back to its beginnings early in the nineteenth century. The reader will find it a rewarding exercise to follow the author’s cue and work backward from the compact, polished presentations in modern textbooks one knows to the original papers. The point is to see the context in which the important ideas arose long before they were honed into the powerful tools over which we dispose nowadays. Sometimes, it can be instructive to set these later refinements aside, just as one can best learn how to orient oneself when hiking in the wilderness by turning off the GPS and relying on an old-fashioned compass and the position of the sun in the sky. Afterwards, one will be free to avail oneself of modern methods with greater discernment. Dieudonné has meticulously reconstructed all the major developments in functional analysis up to the time of his writing in the 1970’s and tells its story well, replete with abundant citations to the primary sources.
What is in the text? The roots of what would come to be known as functional analysis lie in the use of Fourier expansions, as part of which one was led to nascent ideas of spectral theory that were to be enlarged upon by the Sturm-Liouville theory, involving ordinary differential equations of second order in a slightly more general form. The effort after 1880 to apply these ideas in the setting of partial differential equations would stimulate the modern development of spectral theory covered in chapter three. But before going there, Dieudonné swerves into a detour in chapter two on integral, or crypto-integral equations. The original motivation lies in celestial mechanics, where one would like to justify a perturbative approach leading to successive approximations of the true solution in powers of the expansion parameter, presumed to be small. When Cauchy did so in 1835, he arrived at a hierarchy of integral equations, which he showed was convergent, the first time ever that anyone had found a solution to a general type of differential equation (i.e., other than one that can be given in explicit form); see pp. 22-24. Another route leading to integral equations was the potential theory of Green’s. Dieudonné then delves into the convoluted history of the Dirichlet problem during the latter half of the nineteenth century, culminating in Hilbert’s dramatic resolution of the problem in 1899 on the basis of novel direct techniques. Chapter two closes with a presentation of the Beer-Neumann method of constructing solutions to the Dirichlet problem in terms of the boundary via recursive integral equations (later to be called a Fredholm integral equation of second kind).
Chapter three is devoted to the contributions of Schwarz, Weber and Poincaré to the vibrating membrane, the logical extension of the ordinary differential equation for the vibrating string. Although it concerns only a single case, the vibrating membrane served as an ideal testing ground in which to adumbrate the beginnings of spectral theory, which was due to take off with Hilbert’s fundamental paper of 1906. Before turning to the glory days of the first decade of the twentieth century, though, Dieudonné takes a detour in chapter four on the very idea of infinite dimension. He describes in detail the passage from finite-dimensional linear algebra to the infinite-dimensional case. At the hands of Le Roux and Volterra, a general theory of integral equations emerged as a result. It was a good place to encounter the topological issues surrounding the circumstance that there can be several distinct ways for a sequence of functions fₙ to tend to a limit f (for instance, convergence in mean square). Ascoli’s notion of equicontinuity and the concept of an operator that maps a given function into another function arose in this context, but would soon come to be characteristic of the new field of functional analysis as a whole: the relevance of an appropriate choice of topology (what is a trivial issue in finitely many dimensions) and the ubiquitous role of operators on spaces of functions.
Fredholm, Fréchet and Hilbert established the foundations of modern functional analysis; Dieudonné: ‘Between 1900 and 1910, there was a sudden crystallization of all the ideas and methods which had been slowly accumulating during the XIXth century...essentially due to the publication of four fundamental papers: Fredholm’s 1900 paper on integral equations; Lebesgue’s thesis of 1902 on integration; Hilbert’s paper and 1906 on spectral theory and Fréchet’s thesis of 1906 on metric spaces’ (p. 97). The reader will find it eye-opening to follow the path developments took, involving the introduction of such concepts as the Fredholm determinant, resolvent kernels and quadratic forms. Often enough, we can easily recognize these concepts as simple special cases of the more powerful formalism one will have acquired from a course of lectures on modern functional analysis. But this is to look at the history backwards, from our vantage point at its denouement—precisely why the study of the history of mathematics can be such an enjoyable experience! Chapter five closes with an extended reflection on the ‘confluence of geometry, topology and analysis’ thereby conjured up. For Dieudonné, this constellation of ideas is very significant as for the first time, mathematicians began to think in terms of abstract structures (Hausdorff’s axiomatic topology, Fréchet’s metric spaces etc.) and to analyze their properties at the structural level, what calls for a leap in the capacity for abstraction and what would be characteristic of the trend of mathematics all through the following century (especially for the Bourbakians, among whom the author numbers himself), as we see in evident profusion with the rise of category theory and homological algebra after the second world war.
To get a sense of the intellectual excitement involved, consider Cayley’s view of linear algebra as a theory of n-tuples; now, due to the Riesz-Fischer theorem, square-integrable functions can be identified with elements of the space of sequences of numbers (or n-tuples as n tends to infinity). Why Cayley’s concept is inadequate, in general: no such identification is possible for C([a,b]); there, one has to handle continuous functions directly and not through coordinates, as Dieudonné explains (otherwise the Hahn-Banach theorem would be a triviality). At this juncture, functional analysis opens into a wonderful world displaying qualitatively new phenomena which Riesz was among the first to explore, with his concept of the dual space of continuous linear functionals and its potential property of reflexivity. Thus, richer possibilities that defy geometrical intuition fashioned upon finite-dimensional space supervene. The theory of duality becomes subtle and non-trivial as soon as one goes to Banach spaces that are not Hilbert spaces, such as Lᵖ(μ) for p≠2. Another classic contribution is Riesz’ 1918 paper on spectral theory of operators in Banach spaces. Weyl’s theory of singular integral kernels and Carleman kernels represent deep results that probe the intricacies unleashed by going to more general function spaces, not representable in terms of sequences of numbers. Dieudonné’s treatment of these topics is, as usual, generously detailed. For instance, he reconstructs the steps issuing in Hilbert’s discovery of operators having not just pure point but also continuous spectrum (as with the quantum-mechanical model of the hydrogen atom to appear in the 1920’s, where, in addition to the bound states of negative energy, there is a whole continuum of scattering states at every positive value of the energy).
The high point of chapter seven is von Neumann’s clever theory of unbounded operators and their self-adjoint extensions (the need for an analysis of the problem of extension of the domain arises because an unbounded operator can be defined only on a dense subset). It is worth dilating on the comparison of von Neumann to Dirac for a while in order to highlight the former’s vastly greater stature as a mathematician. The latter, as is known, was one of the codiscoverers of the quantum mechanics in its meteoric arrival on the scene in 1924-1925 (along with Heisenberg, Born, Jordan and Schrödinger). Dirac’s version of the theory emphasizes the analogy with the commutative algebra of observable functions on phase space in classical mechanics (c-numbers) and the passage to the non-commutative algebra of observable self-adjoint operators in quantum mechanics (q-numbers), under which the Poisson bracket goes over into the commutator. Dirac also, more than anyone else, taught physicists to think in terms of vectors in Hilbert space (the wavefunction). If one stays at the heuristic level, the basic formalism of quantum mechanics, including scattering theory, stationary perturbation theory and time-dependent perturbation theory, is almost childishly simple. This explains how Born, Heisenberg and Jordan were able to hammer it out in a matter of months in the so-called Dreimännerarbeit [Zeitschrift für Physik 35, 557-615 (1926)]. There is, indeed, almost no real physics in Dirac’s Principles of Quantum Mechanics (for a superior exposition of heuristic quantum mechanics, both in terms of style and in terms of content, see the two volumes of Cohen-Tannoudji; the French gift for clear thought counts for something). Mathematical physicists under the tutelage of Dirac were accustomed to treat operators loosely as if they were matrices with a countable or uncountable basis (take a look at the figures in §3.3 on continuous spectra in the paper just cited). This is scarcely valid from a mathematical point of view. As Dieudonné himself tells us: ‘But if H is not maximal (i.e. both defects are > 0), any hermitian operator which extends H obviously has the same matrix (aₙₘ); and von Neumann showed in great detail how this lack of “one-to-oneness” in the correspondence between matrices and operators led to the weirdest pathology, convincing once for all the analysts that matrices were a totally inadequate tool in spectral theory’ (p. 181). Von Neumann is the one who, all but single-handedly, worked out the rigorous formulation of the quantum mechanics in terms of Hilbert spaces, published in 1932 as the Mathematische Grundlagen der Quantenmechanik.
But von Neumann did not stop there. He went on, in collaboration with Murry, to establish the foundations of the theory of operator algebras, of which Dieudonné remarks, ‘By the wealth and novelty of their techniques and their results, these wonderful papers are certainly the most profound and most difficult which von Neumann ever wrote; they revealed a large number of completely unsuspected phenomena’ (p. 183). The right way of viewing this theory is as a non-commutative generalization of Lebesgue’s measure theory. As Dieudonné observes, it must be seen as a great surprise that von Neumann and Murry could have completed their work years before the elementary concepts of the theory of normed algebras (viz., Banach and C*-algebras) had been elucidated. In the last two sections of chapter seven, Dieudonné covers these and later post-war developments, such as Gelfand’s theorem, the structural theory of Banach algebras and harmonic analysis on locally compact groups, along with the remarkable duality theory that applies to them. The famous names associated with these ideas are those of Frobenius, Weyl, Gelfand and Pontryagin.
Chapter eight concerns locally convex spaces, a generalization of the concept of metric space and about the farthest one can go and still have good theorems. Herein lies the natural setting for Laurent Schwartz’s theory of distributions, which supplies at last a rigorous sense to Dirac’s delta function and proves to be of the utmost value in investigating differential and partial differential equations (to which Dieudonné provides an introduction in chapter nine).
Closing remarks: first, keep in mind that the reader needs to be already fairly conversant with the ideas of functional analysis to profit from this book. Otherwise, one will fail to see the point of much of what Dieudonné has to say. Within space limitations, it is possible only to give precise definitions and a sketch of proof; there is little, if any, discussion of concrete examples beyond the mention that they exist and demonstrate a property or the impossibility of carrying out a certain construction in general. Yet, Dieudonné does perform the invaluable service of delineating a connected narrative of a broad swath of what is by now an extensive discipline and has conveniently assembled references to the primary literature. Compared to what one may be used to from general surveys of the history of mathematics, Dieudonné enters into microscopic detail when sketching the steps along the way. What one can get out of reading him: perspective on how the field of modern functional analysis fits together; what it is important to know and in what order; and a few helpful judgments on quality. In view whereof, warmly to be recommended to the serious and diligent student.
P.S. A minor gripe: why, after the advent of convenient computerized type-setting, could the publisher North-Holland not have invested the seemingly minor sum to have somebody recompose the typewritten manuscript from back in 1981 instead of merely photocopying it? The typewriter font lends the text an unattractive appearance (especially with displayed equations), yet it does not hinder comprehension as much as one might have feared, once one becomes engaged with the material.
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