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Methods of Modern Mathematical Physics #1
Functional Analysis
This book is the first of a multivolume series devoted to an exposition of functional analysis methods in modern mathematical physics. It describes the fundamental principles of functional analysis and is essentially self-contained, although there are occasional references to later volumes. We have included a few applications when we thought that they would provide motivation for the reader. Later volumes describe various advanced topics in functional analysis and give numerous applications in classical physics, modern physics, and partial differential equations.
416 pages, Hardcover
First published January 1, 1972
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April 18, 2026
There once was a time when mathematical physics was exactly what it puts itself out to be, the characterization of the phenomena of nature by means of mathematical reasoning. As standards of rigor improved from loose (17th c.) to the half-hearted (18th c.) to the would-be (19th c.) to perfection (first half of the 20th c.), the meaning of the term changes, as does the nature of the effort demanded of him who would join the scientific community. As is well known, physics does not become mathematical until the second third of the 19th c., with Laplace, Poisson, Cauchy et alia (a notable figure in this connection would be J.F. Fries, who bridges the interim from the 18th c. to the later 19th c.; see our review here).
What is special about the second half of the 20th c. is the equilibrium that prevailed between the necessity to know the basics and the wish to press on to the research frontier. The gifted and aspiring student, in those days, could within a reasonable span of time (to be measured in years) master at least those facets of the perfection of previous decades that may seem potentially relevant to his projected research problem. For instance (among a host of others), A.S. Wightman, J. Glimm, A. Jaffe, E. Lieb and so forth. These men and women learned their mathematical prerequisites straight from the journal literature. We lesser ones of the succeeding generations could scarcely hope to do so, however. Therefore, the authors of the present work, Michael Reed and Barry Simon, two among the great ones during the era of mathematical physics’ flourishing, condescended (in the proper, almost theological sense) to write Methods of Modern Mathematical Physics in four volumes, published starting in 1980 by Academic Press.
Vol. I, Functional Analysis: Revised and Enlarged Edition (1980) covers the elements of functional analysis, from the beginner’s level all the way to the rather advanced material on locally convex spaces needed for quantum field theory and other applications. Vol. I comprises Chaps. I-VIII (Chap. IX from Vol. II on the (distributional) Fourier transform has, in the present edition, been appended as a supplement for the convenience of students who could not afford to purchase all of Vol. II). Even the preliminaries in Chap. 1 can be, or were so at least, exciting to the neophyte who will be seeing ε-δ arguments for the first time in the advanced setting of metric and normed linear spaces that goes, in principle, well beyond the calculus. See for instance §5 on two convergence arguments (the so-called diagonal trick happens to be far trickier than one might suspect, or at least than this reviewer did and does every time he comes across it in a problem).
Chap. II sets forth the basic theory of Hilbert space geometry (which is to be distinguished from the theory of operators acting in a Hilibert space, Chaps. VI-VIII below). The Riesz lemma (what provides the rationale for Dirac’s bra-ket notation) and such things will be new to the mere student of quantum mechanics, and so will be the tensor product and the fine §5 on ergodic theory, which leads in just seven pages to proofs of von Neumann’s and Birkhoff’s ergodic theorems.
So far, up to Chap. III on Banach spaces and maybe through Chap. IV on topological spaces, we remain on the same track alongside the pure mathematical analyst, namely, in the traditional loci of real analysis and beginning functional analysis. Yet, Chap. V on locally convex spaces (of Fréchet or of rapid decrease type) jumps beyond what most physicists and even many pure mathematicians know (it will not be until Vol. II that the reader finds out why one might want to articulate notions such as these). The Leray-Schauder-Tychonoff and the Markov-Kakutani fixed point theorems are key; the former, for instance, suffices to show existence of local solutions to ordinary differential equations rather more elegantly than via Picard’s method of iterations (on the latter, see E.A. Coddington’s An Introduction to Ordinary Differential Equations (Dover, 1989); our review here).
Chaps. VI and VII on bounded operators in Banach spaces and their spectral theory return to more familiar ground, albeit succinctly. Those wanting more substance may turn to B.D. MacCluer’s Elementary Functional Analysis (Springer, 2009). Yet Reed and Simon’s objective here is to convey what is necessary in order to go on the unbounded case, hence, to dwell on arcana concerning the various possible notions of convergence (norm, weak, strong, weak*). Thus, Chap. VIII on unbounded operators and the spectral theorem in the unbounded case forms the heart of this book, and will not disappoint. (Also, indispensable for Vol. II, which we intend to review in a moment.)
Let us say something about the mathematical physics made possible by the work covered in Vol. I. If the reader wishes to find out a lot more on this subject, he may consult the extensive chapter-end notes appended by Reed and Simon themselves. The founders of the quantum mechanics and quantum field theory neglect to show the correspondence between their theoretical model and experimental reality (to the extent the former can be defined, at all). If one is to do so, the tools developed by mathematical physicists and expounded in this textbook are essential. For instance, if quantum mechanics is to obey a temporal dependence of the form of a 1-parameter group of unitary operators, the Hamiltonian must be self-adjoint. Self-adjointness is presumed by J. von Neumann in 1932 [see his Mathematische Grundlagen der Quantenmechanik, 1932; our review here] but not proved until 1951, when T. Kato applies F. Rellich’s theorem to atomic systems [cf. F. Rellich, Störungstheorie der Spektralzerlegung, II, Math. Ann. 116, 555-570 (1939) and T. Kato, Fundamental properties of Hamiltonian operators of Schrödinger type, Trans. Amer. Math. Soc. 70, 195-211 (1951)]. Similar remarks would apply to the programs of constructive quantum field theory and of atomic physics (perhaps even condensed-matter physics). If it were not for the mathematical physicists, we should fail of knowledge.
A little critique of Zaslow-type physico-mathematics of recent decades: to be sharp yet balanced, one may concede that physico-mathematics can lead to interesting theoretical structures to explore, just demur, on the other hand, that it tends to disconnect from reality. [See Arthur Jaffe and Frank Quinn, ‘Theoretical mathematics’: Toward a cultural synthesis of mathematics and theoretical physics, Bulletin of the American Mathematical Society 29(1), 1-13 (1993). (At a seminar the author of this review once gave as a candidate to Jaffe’s group at Harvard, E. Zaslow recognized a not-quite-trivial claim and was kindhearted enough to ask me to prove it on the spot at the blackboard using Čech cohomology.)]
Five stars: outstanding clarity throughout, well stocked with examples, illustrative of typical cases as well as of pathologies (as in the contraction resp. expansion of the spectrum upon perturbation). As to Reed and Simon’s homework exercises: do not be dismayed by how numerous they are. What usually happens—one must suppose, but does anyone ever compare notes in order to check?—is that somewhere between 2/3 and 4/5 of the problems will strike any given student as fairly easy, while the others, as rather hard (though not the same ones for different people!). Moreover, furnished with excellent discursive chapter-end notes containing numerous references to the original literature.
What is special about the second half of the 20th c. is the equilibrium that prevailed between the necessity to know the basics and the wish to press on to the research frontier. The gifted and aspiring student, in those days, could within a reasonable span of time (to be measured in years) master at least those facets of the perfection of previous decades that may seem potentially relevant to his projected research problem. For instance (among a host of others), A.S. Wightman, J. Glimm, A. Jaffe, E. Lieb and so forth. These men and women learned their mathematical prerequisites straight from the journal literature. We lesser ones of the succeeding generations could scarcely hope to do so, however. Therefore, the authors of the present work, Michael Reed and Barry Simon, two among the great ones during the era of mathematical physics’ flourishing, condescended (in the proper, almost theological sense) to write Methods of Modern Mathematical Physics in four volumes, published starting in 1980 by Academic Press.
Vol. I, Functional Analysis: Revised and Enlarged Edition (1980) covers the elements of functional analysis, from the beginner’s level all the way to the rather advanced material on locally convex spaces needed for quantum field theory and other applications. Vol. I comprises Chaps. I-VIII (Chap. IX from Vol. II on the (distributional) Fourier transform has, in the present edition, been appended as a supplement for the convenience of students who could not afford to purchase all of Vol. II). Even the preliminaries in Chap. 1 can be, or were so at least, exciting to the neophyte who will be seeing ε-δ arguments for the first time in the advanced setting of metric and normed linear spaces that goes, in principle, well beyond the calculus. See for instance §5 on two convergence arguments (the so-called diagonal trick happens to be far trickier than one might suspect, or at least than this reviewer did and does every time he comes across it in a problem).
Chap. II sets forth the basic theory of Hilbert space geometry (which is to be distinguished from the theory of operators acting in a Hilibert space, Chaps. VI-VIII below). The Riesz lemma (what provides the rationale for Dirac’s bra-ket notation) and such things will be new to the mere student of quantum mechanics, and so will be the tensor product and the fine §5 on ergodic theory, which leads in just seven pages to proofs of von Neumann’s and Birkhoff’s ergodic theorems.
So far, up to Chap. III on Banach spaces and maybe through Chap. IV on topological spaces, we remain on the same track alongside the pure mathematical analyst, namely, in the traditional loci of real analysis and beginning functional analysis. Yet, Chap. V on locally convex spaces (of Fréchet or of rapid decrease type) jumps beyond what most physicists and even many pure mathematicians know (it will not be until Vol. II that the reader finds out why one might want to articulate notions such as these). The Leray-Schauder-Tychonoff and the Markov-Kakutani fixed point theorems are key; the former, for instance, suffices to show existence of local solutions to ordinary differential equations rather more elegantly than via Picard’s method of iterations (on the latter, see E.A. Coddington’s An Introduction to Ordinary Differential Equations (Dover, 1989); our review here).
Chaps. VI and VII on bounded operators in Banach spaces and their spectral theory return to more familiar ground, albeit succinctly. Those wanting more substance may turn to B.D. MacCluer’s Elementary Functional Analysis (Springer, 2009). Yet Reed and Simon’s objective here is to convey what is necessary in order to go on the unbounded case, hence, to dwell on arcana concerning the various possible notions of convergence (norm, weak, strong, weak*). Thus, Chap. VIII on unbounded operators and the spectral theorem in the unbounded case forms the heart of this book, and will not disappoint. (Also, indispensable for Vol. II, which we intend to review in a moment.)
Let us say something about the mathematical physics made possible by the work covered in Vol. I. If the reader wishes to find out a lot more on this subject, he may consult the extensive chapter-end notes appended by Reed and Simon themselves. The founders of the quantum mechanics and quantum field theory neglect to show the correspondence between their theoretical model and experimental reality (to the extent the former can be defined, at all). If one is to do so, the tools developed by mathematical physicists and expounded in this textbook are essential. For instance, if quantum mechanics is to obey a temporal dependence of the form of a 1-parameter group of unitary operators, the Hamiltonian must be self-adjoint. Self-adjointness is presumed by J. von Neumann in 1932 [see his Mathematische Grundlagen der Quantenmechanik, 1932; our review here] but not proved until 1951, when T. Kato applies F. Rellich’s theorem to atomic systems [cf. F. Rellich, Störungstheorie der Spektralzerlegung, II, Math. Ann. 116, 555-570 (1939) and T. Kato, Fundamental properties of Hamiltonian operators of Schrödinger type, Trans. Amer. Math. Soc. 70, 195-211 (1951)]. Similar remarks would apply to the programs of constructive quantum field theory and of atomic physics (perhaps even condensed-matter physics). If it were not for the mathematical physicists, we should fail of knowledge.
A little critique of Zaslow-type physico-mathematics of recent decades: to be sharp yet balanced, one may concede that physico-mathematics can lead to interesting theoretical structures to explore, just demur, on the other hand, that it tends to disconnect from reality. [See Arthur Jaffe and Frank Quinn, ‘Theoretical mathematics’: Toward a cultural synthesis of mathematics and theoretical physics, Bulletin of the American Mathematical Society 29(1), 1-13 (1993). (At a seminar the author of this review once gave as a candidate to Jaffe’s group at Harvard, E. Zaslow recognized a not-quite-trivial claim and was kindhearted enough to ask me to prove it on the spot at the blackboard using Čech cohomology.)]
Five stars: outstanding clarity throughout, well stocked with examples, illustrative of typical cases as well as of pathologies (as in the contraction resp. expansion of the spectrum upon perturbation). As to Reed and Simon’s homework exercises: do not be dismayed by how numerous they are. What usually happens—one must suppose, but does anyone ever compare notes in order to check?—is that somewhere between 2/3 and 4/5 of the problems will strike any given student as fairly easy, while the others, as rather hard (though not the same ones for different people!). Moreover, furnished with excellent discursive chapter-end notes containing numerous references to the original literature.
February 22, 2024
Writing a math book is an art form: Reed and Simon are the Michelangelo of said form.
Want to Read
November 12, 2023Aparece como ejemplo la función 1/x renormalizada.
((1/x)_+ renormalized)
Es posible interpretar 1/x como una distribución
((1/x)_+ renormalized)
Es posible interpretar 1/x como una distribución
February 6, 2017
Most theorems are rigorously proved, and although the book becomes more and more biased towards mathematical physics (i.e., methods for proving self-adjointness, analysis of spectra and scattering theory, as stated in the section "Three Mathematical Problems in Quantum Mechanics".
Want to Read
January 27, 2019Analysis and its applications to quantum physics
Displaying 1 - 5 of 5 reviews