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Graduate Texts in Mathematics #253
Elementary Functional Analysis
Functional analysis arose in the early twentieth century and gradually, conquering one stronghold after another, became a nearly universal mathematical doctrine, not merely a new area of mathematics, but a new mathematical world view. Its appearance was the inevitable consequence of the evolution of all of nineteenth-century mathematics, in particular classical analysis and mathematical physics. Its original basis was formed by Cantor’s theory of sets and linear algebra. Its existence answered the question of how to state general principles of a broadly interpreted analysis in a way suitable for the most diverse situations. A.M. Vershik ([45], p. 438). This text evolved from the content of a one semester introductory course in fu- tional analysis that I have taught a number of times since 1996 at the University of Virginia. My students have included ?rst and second year graduate students prep- ing for thesis work in analysis, algebra, or topology, graduate students in various departments in the School of Engineering and Applied Science, and several und- graduate mathematics or physics majors. After a ?rst draft of the manuscript was completed, it was also used for an independent reading course for several und- graduates preparing for graduate school.
218 pages, Hardcover
First published January 1, 2008
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July 25, 2020
There is no lack of introductory textbooks on functional analysis. Why would we need another? To this reviewer’s knowledge, none has established itself as the canonical choice in the mathematical curriculum. His first exposure to functional analysis was from the first volume of Reed and Simon’s monumental Methods of Modern Mathematical Physics. Despite the title and despite the authors being physicists, this first volume is a good presentation of pure mathematics, equipped with fairly extensive and helpful chapter-end notes that go into how the concepts apply to physics (the second through fourth volumes of the series are concerned with more specialized topics that arise from, and properly pertain to, mathematical physics). There are others, but none really stands out among the ones that merit mention; Kolmogorov’s is too elementary and doesn’t get very far into the subject; Edwards, though comprehensive and perhaps serviceable, is too old and dated; Rudin’s textbook on functional analysis is, unfortunately, pitched at much too high a level for the beginning student; Berberian’s a little to narrow in scope etc. Thus, it is not beyond the bounds of plausibility that a new entrant to the field could prove itself worthy of canonical status. This review is devoted to the recent volume by Barbara MacCluer in the Springer series of graduate texts in mathematics, entitled Elementary Functional Analysis. MacCluer is an expert in operator theory and has the teaching experience to attempt this work, which she describes as aimed at the beginning graduate student, perhaps in engineering or physics, but also one who wishes to do doctoral work in analysis, algebra or topology. She intentionally keeps the prerequisites to a minimum, nothing more than an undergraduate preparation in real analysis, linear algebra, point-set topology and complex analysis (although she downplays the importance of the last-named, this reviewer thinks it necessary to internalize the material).
As we shall see, however, to call it elementary is something of a misnomer. While concise, at barely two hundred pages, the brevity is not obtained at the cost of sufficient depth and detail; the topics covered are treated in full and the overall length of the text is kept under control through a careful selection of topics (and omission thereof).
MacCluer starts out with elementary Banach and Hilbert space theory, emphasizing their geometrical properties. At the end of this brief first chapter, she employs complex analysis to give a non-trivial example of a Hilbert space, viz., the Bergman space of square-integrable analytic functions on the unit disk—which illustrates what was said above, to the effect that a good understanding of complex analysis is necessary for a proper appreciation of her text, despite her protestations to the contrary. It also raises what for this reviewer has been a long-standing puzzle: if two Hilbert spaces are isomorphic whenever their dimensions have the same cardinality, what does one gain by ever thinking about a Hilbert space defined in concrete terms rather than merely as the abstract span of an orthonormal set of the requisite cardinality? Yet it certainly appears that one does profit by thinking so. Consider, for instance, elementary quantum mechanics. Anyone who has solved the problem of the hydrogen atom will have been exposed to pictures of its low-lying eigenfunctions and, to be sure, such pictures do very much aid intuition, as is the case in organic chemistry with Pauling’s theory of hybridization of s and p atomic orbitals in terms of which bonding in small polyatomic molecules can be so nicely explicated. Abstract theory (as a corollary to the Reisz-Fischer theorem), however, tells us that the space of square-integrable wavefunctions in configuration space is isomorphic to the space of square-summable sequences, in which any spatial intuition is entirely lacking. The lesson: a great deal of semantic value hides within the said isomorphism to l2(N). One has essentially to solve the physics of the problem before one can set up the relevant isomorphism.
Back to MacCluer: chapter two covers the basics of operator theory, through the concept of adjoint for bounded operators in Hilbert or Banach space. The treatment is furnished with a wealth of examples: shift operators, multiplication operators, integral operators and the Fourier transform for periodic functions. As we see, even though the text is billed as ‘elementary’, the author is willing to venture into more advanced territory than one might expect from an exposition for rank beginners. The third chapter sets forth the big three foundational results in functional analysis, the Hahn-Banach theorem, the principle of uniform boundedness and the open mapping theorem (along with the closed graph theorem and the closely related Banach-Steinhaus theorem). MacCluer’s proofs are methodical and easy to follow for a beginner, broken down into a series of lemmas and corollaries. Moreover, the ideas are illustrated with a few well-chosen examples.
The best part of this text, in this reviewer’s opinion, can be found the subsequent two chapters in which MacCluer’s credentials as an operator theorist shine. Beginning with compact operators (which lend themselves particularly well to a structural analysis), we get a miniature course in spectral theory, invariant subspaces and the Fredholm alternative, followed by a masterful synopsis of the basic theory of Banach and C* algebras in chapter five. The various parts of the spectrum of a bounded operator (pure point, continuous and residual) and their significance in terms of the kernel and range of the resolvent are spelled out very clearly and supported with worked examples, in which the pieces of the spectrum for some toy operators can be explicitly determined. MacCluer’s discussion of ideals and homomorphisms of Banach algebras, commutative Banach algebras, weak topologies, the Banach-Alaoglu theorem, the Gelfand transform and the continuous functional calculus for normal operators is nothing short of excellent. It is amazing, as well, that she manages to pack so much material into one chapter in a concise textbook, numbering just 207 pages in total. Lastly, the treatment of the spectral theorem for bounded normal operators in separable Hilbert spaces in chapter six is very methodical and orderly. The basic concepts such as the spectral measure and steps in the proof, starting with cyclic vectors and building up to the general case, are laid out very clearly, and at the close one does have an intuitive feeling for why the somewhat surprising, at first glance, spectral theorem is indeed true.
In this reviewer’s view, the heart of MacCluer’s text is to be found in the numerous well-chosen homework exercises, some 192 in all over six chapters. Their level is pitched about right, not too easy but also not too hard; a reasonably enterprising student should be able to get most, if not all, of them (some of the more difficult ones are given hints). The exercises round out the material in the chapters nicely, involving the student enough to give him a sufficient command of the subject. MacCluer’s penchant for operator theory is on display here, and one encounters the definitions and basic properties, at least, of, among other things, Toeplitz operators, weighted shift operators, the Banach limit and frames (used in wavelet theory for image processing). It is a pleasure, say, to work out the pieces of the spectrum for some simple example operators that can be defined in closed form. It is not a fault of the author’s in a text of such delimited scope, but one wishes one could see some significant applications of these ideas, of which one is given a glimpse in the exercises.
All around, MacCluer’s effort can be judged a success and perhaps this course will gain the recognition it deserves as a fine introduction to the subject of functional analysis. As the overview above shows, the author accomplishes her stated ‘desire to start with the basics and still travel a path through some significant part of modern functional analysis’. This is a very personal objective; one doubts whether another elementary textbook exists that manages such a feat, let alone attempts it. Another personal note is the anecdotes on the mathematicians who originally developed the theory and commentary on their history she weaves into the text, something not ordinarily found in textbooks at this level but pleasant for the reader nonetheless. Moreover, MacCluer succeeds well in her wish to write an accessible text for the reader who is learning on his own, without the benefit of a formal course or instructor. The reader’s appetite to delve more deeply into the mysteries of functional analysis will be whetted!
As we shall see, however, to call it elementary is something of a misnomer. While concise, at barely two hundred pages, the brevity is not obtained at the cost of sufficient depth and detail; the topics covered are treated in full and the overall length of the text is kept under control through a careful selection of topics (and omission thereof).
MacCluer starts out with elementary Banach and Hilbert space theory, emphasizing their geometrical properties. At the end of this brief first chapter, she employs complex analysis to give a non-trivial example of a Hilbert space, viz., the Bergman space of square-integrable analytic functions on the unit disk—which illustrates what was said above, to the effect that a good understanding of complex analysis is necessary for a proper appreciation of her text, despite her protestations to the contrary. It also raises what for this reviewer has been a long-standing puzzle: if two Hilbert spaces are isomorphic whenever their dimensions have the same cardinality, what does one gain by ever thinking about a Hilbert space defined in concrete terms rather than merely as the abstract span of an orthonormal set of the requisite cardinality? Yet it certainly appears that one does profit by thinking so. Consider, for instance, elementary quantum mechanics. Anyone who has solved the problem of the hydrogen atom will have been exposed to pictures of its low-lying eigenfunctions and, to be sure, such pictures do very much aid intuition, as is the case in organic chemistry with Pauling’s theory of hybridization of s and p atomic orbitals in terms of which bonding in small polyatomic molecules can be so nicely explicated. Abstract theory (as a corollary to the Reisz-Fischer theorem), however, tells us that the space of square-integrable wavefunctions in configuration space is isomorphic to the space of square-summable sequences, in which any spatial intuition is entirely lacking. The lesson: a great deal of semantic value hides within the said isomorphism to l2(N). One has essentially to solve the physics of the problem before one can set up the relevant isomorphism.
Back to MacCluer: chapter two covers the basics of operator theory, through the concept of adjoint for bounded operators in Hilbert or Banach space. The treatment is furnished with a wealth of examples: shift operators, multiplication operators, integral operators and the Fourier transform for periodic functions. As we see, even though the text is billed as ‘elementary’, the author is willing to venture into more advanced territory than one might expect from an exposition for rank beginners. The third chapter sets forth the big three foundational results in functional analysis, the Hahn-Banach theorem, the principle of uniform boundedness and the open mapping theorem (along with the closed graph theorem and the closely related Banach-Steinhaus theorem). MacCluer’s proofs are methodical and easy to follow for a beginner, broken down into a series of lemmas and corollaries. Moreover, the ideas are illustrated with a few well-chosen examples.
The best part of this text, in this reviewer’s opinion, can be found the subsequent two chapters in which MacCluer’s credentials as an operator theorist shine. Beginning with compact operators (which lend themselves particularly well to a structural analysis), we get a miniature course in spectral theory, invariant subspaces and the Fredholm alternative, followed by a masterful synopsis of the basic theory of Banach and C* algebras in chapter five. The various parts of the spectrum of a bounded operator (pure point, continuous and residual) and their significance in terms of the kernel and range of the resolvent are spelled out very clearly and supported with worked examples, in which the pieces of the spectrum for some toy operators can be explicitly determined. MacCluer’s discussion of ideals and homomorphisms of Banach algebras, commutative Banach algebras, weak topologies, the Banach-Alaoglu theorem, the Gelfand transform and the continuous functional calculus for normal operators is nothing short of excellent. It is amazing, as well, that she manages to pack so much material into one chapter in a concise textbook, numbering just 207 pages in total. Lastly, the treatment of the spectral theorem for bounded normal operators in separable Hilbert spaces in chapter six is very methodical and orderly. The basic concepts such as the spectral measure and steps in the proof, starting with cyclic vectors and building up to the general case, are laid out very clearly, and at the close one does have an intuitive feeling for why the somewhat surprising, at first glance, spectral theorem is indeed true.
In this reviewer’s view, the heart of MacCluer’s text is to be found in the numerous well-chosen homework exercises, some 192 in all over six chapters. Their level is pitched about right, not too easy but also not too hard; a reasonably enterprising student should be able to get most, if not all, of them (some of the more difficult ones are given hints). The exercises round out the material in the chapters nicely, involving the student enough to give him a sufficient command of the subject. MacCluer’s penchant for operator theory is on display here, and one encounters the definitions and basic properties, at least, of, among other things, Toeplitz operators, weighted shift operators, the Banach limit and frames (used in wavelet theory for image processing). It is a pleasure, say, to work out the pieces of the spectrum for some simple example operators that can be defined in closed form. It is not a fault of the author’s in a text of such delimited scope, but one wishes one could see some significant applications of these ideas, of which one is given a glimpse in the exercises.
All around, MacCluer’s effort can be judged a success and perhaps this course will gain the recognition it deserves as a fine introduction to the subject of functional analysis. As the overview above shows, the author accomplishes her stated ‘desire to start with the basics and still travel a path through some significant part of modern functional analysis’. This is a very personal objective; one doubts whether another elementary textbook exists that manages such a feat, let alone attempts it. Another personal note is the anecdotes on the mathematicians who originally developed the theory and commentary on their history she weaves into the text, something not ordinarily found in textbooks at this level but pleasant for the reader nonetheless. Moreover, MacCluer succeeds well in her wish to write an accessible text for the reader who is learning on his own, without the benefit of a formal course or instructor. The reader’s appetite to delve more deeply into the mysteries of functional analysis will be whetted!
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