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Real Analysis: Modern Techniques and Their Applications
An in-depth look at real analysis and its applications-now expanded and revised.
This new edition of the widely used analysis book continues to cover real analysis in greater detail and at a more advanced level than most books on the subject. Encompassing several subjects that underlie much of modern analysis, the book focuses on measure and integration theory, point set topology, and the basics of functional analysis. It illustrates the use of the general theories and introduces readers to other branches of analysis such as Fourier analysis, distribution theory, and probability theory.
This edition is bolstered in content as well as in scope-extending its usefulness to students outside of pure analysis as well as those interested in dynamical systems. The numerous exercises, extensive bibliography, and review chapter on sets and metric spaces make Real Analysis: Modern Techniques and Their Applications, Second Edition invaluable for students in graduate-level analysis courses. New features include:
* Revised material on the n-dimensional Lebesgue integral.
* An improved proof of Tychonoff's theorem.
* Expanded material on Fourier analysis.
* A newly written chapter devoted to distributions and differential equations.
* Updated material on Hausdorff dimension and fractal dimension.
This new edition of the widely used analysis book continues to cover real analysis in greater detail and at a more advanced level than most books on the subject. Encompassing several subjects that underlie much of modern analysis, the book focuses on measure and integration theory, point set topology, and the basics of functional analysis. It illustrates the use of the general theories and introduces readers to other branches of analysis such as Fourier analysis, distribution theory, and probability theory.
This edition is bolstered in content as well as in scope-extending its usefulness to students outside of pure analysis as well as those interested in dynamical systems. The numerous exercises, extensive bibliography, and review chapter on sets and metric spaces make Real Analysis: Modern Techniques and Their Applications, Second Edition invaluable for students in graduate-level analysis courses. New features include:
* Revised material on the n-dimensional Lebesgue integral.
* An improved proof of Tychonoff's theorem.
* Expanded material on Fourier analysis.
* A newly written chapter devoted to distributions and differential equations.
* Updated material on Hausdorff dimension and fractal dimension.
416 pages, Hardcover
First published September 24, 1984
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Displaying 1 - 9 of 9 reviews
May 1, 2024
Gerald B. Folland’s Real Analysis: Modern Techniques and their Applications (Wiley Interscience, second edition, 1999) proves to be ideal for a second course on real analysis in which the aim should be to solidify one’s comprehension of concepts to which one will have been exposed in a first course, with the objective of attaining real facility with and mastery of the subject. At the end of the day, beginning real analysis isn’t that hard but does constitute an essential tool for scaling the lofty peaks of functional analysis. This reviewer, however, is not very familiar as yet with advanced topics in real analysis such as the Poisson integral or Hardy spaces to be found in papa Rudin so it would be premature for him to say whether Folland is a good preparation for them or not. The time for that will come!
On the edition itself: unfortunately, poorly typeset (intersperses Times Roman for the text with Computer Modern for the formulae) and seems to have excessively many typos (see the six-page corrigendum at the author’s website). It has been twenty-five years since the second edition came out, so we may conclude that Wiley-Interscience isn’t that interested in printing a clean and corrected version even though with computer typesetting it should be inexpensive to do so.
The elements of measure theory go by quickly in chapter one but, for this reviewer, it raises an interesting question as to whether the prevailing concept of measure can be expanded in a manner that would prove productive. For instance, measure theory is a touchy subject in infinitely many dimensions, but even in finitely many dimensions one might wonder whether relaxing the condition of additivity to subadditivity could embrace a wealth of potential applications, say in game theory, where there could be cooperative or non-local effects going under the heading of good will (to employ a term from accounting) that are not adequately captured by our usual notions of a utility function. Just a speculation for the time being, though!
The exposition is exceptionally clear. Unlike many top mathematicians who, in their vaulting pride, would like to keep their mathematics an esoteric arcanum rather than be open and clear, Folland is willing to point out things that may be trivial to derive but aid comprehension; e.g. on why outer measure is a useful concept and immediately thereafter, why the measurability condition has to be imposed in the first place. Folland’s style is characterized by good logical flow and an efficient conceptual scheme (for instance, he will define a Borel measure on the real line for any distribution function F and obtain the Lebesgue measure from F(x)=x; then he enters into a rapid discussion of regularity which Royden defers to much later; Theorem 1.19 means that every measurable set is approximable by something nice).
Nothing in chapter one is deep but one should do the exercises to get familiar with the typical tricks. The chapter-end notes adopt a somewhat more advanced perspective than Royden’s (e.g., Folland comments on the connection between σ-algebras and countable ordinals using transfinite induction).
Chapters two and three offer a good and concise coverage of the Lebesgue integral and allied topics, such as the Radon-Nikodym derivative. Folland, as usual, helps the exposition along with occasional insightful, if in principle incidental comments, such as where he points out nicely why the dominated convergence theorem works or, later on, outlines a good explanation of why the Lebesgue integral is more versatile than the Riemann integral. The pace is brisk. The first three chapters spanning just a hundred pages are equivalent to almost half of Royden! At the close of chapter two, one will find again a fine specimen of how Folland’s chapter-end notes are useful, particularly in the discussion of extended notions of integration (leading to that of Henstock-Kurzweil) and why the Lebesgue integral is normally enough.
The treatment of point-set topology in chapter four is routine for anyone who’s had analysis before – albeit very much an analyst’s take! Topology traditionally falls into two domains, of low respectively high dimensionality. Now, the low-dimensional side is what one typically encounters in popular literature (genus of coffee cups and doughnuts and so forth), but the high-dimensional side becomes crucial if one wants to get serious about functional analysis. Folland’s treatment is telegraphic considering the range of topics, with little space devoted to each, just the minimum subsequently to be needed for analysis. Yet one will be pleased with the efficient proofs of Urysohn’s lemma and the Tietze extension theorem (if one has never done it oneself before, as would be the case when working through Royden).
The functional analysis itself starting in chapter five starts to be rather interesting, for anyone who cares to go beyond a superficial appropriation of the subject: theory of linear functionals under different conditions (in X*, X**, X***, where Folland explains well the various topologies: norm, weak, weak* etc.) and the Baire category theorem along with its consequences such as the open mapping theorem, the closed graph theorem and the principle of uniform boundedness. The sections of this chapter are outfitted with plenty of good problems; for instance, the reader is asked to show that the set of nowhere differentiable functions is residual in C([0,1]). It proves frustrating, however, to dip into topological vector spaces without any significant applications exhibited. Chapter six on the basic theory of L^p spaces and duality or interpolation therein, really just a coda to chapter five, does offer some satisfying exercises; for instance, to prove necessary and sufficient conditions for L^p to be contained in L^q when p < q or q < p.
In general, Folland’s homework problems are for the most part routine and never go beyond a moderate level of difficulty. Almost always hints are provided if the problem is non-trivial. Often, the problems are intended to illustrate the theory through examples or counterexamples, but therefore do not call upon very extensive resources. It should be enough to flip through the previous section to find the relevant theorems that apply under the special case at hand. In particular, the numerous problems in chapter four on point-set topology are all straightforward.
What often happens, though, is that even if the problem itself is not intrinsically hard one can get stuck on trying a method of proof that will not work. For instance, to show that a linear operator T is closed one does not have to prove directly that for every xₙ → x, Txₙ → Tx; rather, that if xₙ → x and Txₙ → y, then y = Tx (not the same thing!). This reviewer remembers, in contrast, that in Royden’s standard beginning graduate-level textbook on real analysis (reviewed by us here), although it has a fair amount of overlap in the problems with Folland, often includes at least a few substantially challenging exercises in each chapter. A quick comparison of Folland with Royden, for the record: as to pedagogical style, Folland starts with measure and integral over arbitrary measure spaces rather than use R as a jumping-off point, as does Royden (which necessitates repetition in the later chapters on general measure spaces). Second, Folland seems to address himself to a reader who can be expected to want a more advanced point of view, and to be prepared for it.
But the choice between Folland and Royden isn’t obvious. Royden might be more thorough while Folland may give a little more advanced point of view on what he does cover. On the other hand, Folland’s homework exercises are by and large too easy, especially if one avails oneself of the hints. Suggested resolution: go through Royden first to learn the basics of the subject, then follow with Folland sometime later to refine one’s understanding (this way one can skip the standard exercises one has already done and focus on the interesting ones, of which there are many particularly in chapter five onwards; for instance, to derive Bochner’s integral for vector-valued functions and applications of Baire category including the principle of condensation of singularities).
Let us comment now on the theory of Radon measures in chapter seven. The introductory paragraph illustrates once again how Folland likes to provide helpful perspective on what he is going to do. The rather technical definition of a Radon measure shows how, back in the early decades of the twentieth century during the initial flourishing of the discipline, one has already worked all the fine functional-analytic details, issuing in a theory impressive for its scope and consequence. For instance, the inner and outer regularity properties of measures are surveyed under different conditions (σ-finiteness or σ-compactness or second countability etc.), the Riesz representation theorem and Lusin’s theorem are proved, and a good duality theory exists as well. Also, it deserves notice that the mere realization that locally compact Hausdorff spaces are about as general as one can get and still have an interesting theory of integration is itself significant. The whole theory in this chapter is a nice application of all the concepts in measure theory, point-set topology and functional analysis patiently built up in the preceding chapters, indicative of the depth of the original ideas.
This review covers the text only through chapter seven, not counting the applications in chapters eight through eleven. Arguably, it may be better to learn Fourier analysis, distribution theory and probability theory elsewhere. The main drawback of stopping here is that one has yet to get to any deep employment of all the ideas one has seen so painstakingly developed. But an education is not a matter of a day and will go on for the rest of one’s life, so we hope eventually to report on that.
In closing let us remark on the pleasure of doing real analysis in contrast with complex analysis. In the absence of the quite strong condition of holomorphicity to rely on, its results may be less elegant or ‘pretty’, but for that, also sturdier or manlier. Thus, get down to the grindstone with Folland if one ever wishes to know real analysis in order to prove something original in mathematical physics (not that is, after the manner of the field as we have it today, where heuristics and conjectures suffice, but in the old-school style where one was supposed to justify one’s contentions)!
On the edition itself: unfortunately, poorly typeset (intersperses Times Roman for the text with Computer Modern for the formulae) and seems to have excessively many typos (see the six-page corrigendum at the author’s website). It has been twenty-five years since the second edition came out, so we may conclude that Wiley-Interscience isn’t that interested in printing a clean and corrected version even though with computer typesetting it should be inexpensive to do so.
The elements of measure theory go by quickly in chapter one but, for this reviewer, it raises an interesting question as to whether the prevailing concept of measure can be expanded in a manner that would prove productive. For instance, measure theory is a touchy subject in infinitely many dimensions, but even in finitely many dimensions one might wonder whether relaxing the condition of additivity to subadditivity could embrace a wealth of potential applications, say in game theory, where there could be cooperative or non-local effects going under the heading of good will (to employ a term from accounting) that are not adequately captured by our usual notions of a utility function. Just a speculation for the time being, though!
The exposition is exceptionally clear. Unlike many top mathematicians who, in their vaulting pride, would like to keep their mathematics an esoteric arcanum rather than be open and clear, Folland is willing to point out things that may be trivial to derive but aid comprehension; e.g. on why outer measure is a useful concept and immediately thereafter, why the measurability condition has to be imposed in the first place. Folland’s style is characterized by good logical flow and an efficient conceptual scheme (for instance, he will define a Borel measure on the real line for any distribution function F and obtain the Lebesgue measure from F(x)=x; then he enters into a rapid discussion of regularity which Royden defers to much later; Theorem 1.19 means that every measurable set is approximable by something nice).
Nothing in chapter one is deep but one should do the exercises to get familiar with the typical tricks. The chapter-end notes adopt a somewhat more advanced perspective than Royden’s (e.g., Folland comments on the connection between σ-algebras and countable ordinals using transfinite induction).
Chapters two and three offer a good and concise coverage of the Lebesgue integral and allied topics, such as the Radon-Nikodym derivative. Folland, as usual, helps the exposition along with occasional insightful, if in principle incidental comments, such as where he points out nicely why the dominated convergence theorem works or, later on, outlines a good explanation of why the Lebesgue integral is more versatile than the Riemann integral. The pace is brisk. The first three chapters spanning just a hundred pages are equivalent to almost half of Royden! At the close of chapter two, one will find again a fine specimen of how Folland’s chapter-end notes are useful, particularly in the discussion of extended notions of integration (leading to that of Henstock-Kurzweil) and why the Lebesgue integral is normally enough.
The treatment of point-set topology in chapter four is routine for anyone who’s had analysis before – albeit very much an analyst’s take! Topology traditionally falls into two domains, of low respectively high dimensionality. Now, the low-dimensional side is what one typically encounters in popular literature (genus of coffee cups and doughnuts and so forth), but the high-dimensional side becomes crucial if one wants to get serious about functional analysis. Folland’s treatment is telegraphic considering the range of topics, with little space devoted to each, just the minimum subsequently to be needed for analysis. Yet one will be pleased with the efficient proofs of Urysohn’s lemma and the Tietze extension theorem (if one has never done it oneself before, as would be the case when working through Royden).
The functional analysis itself starting in chapter five starts to be rather interesting, for anyone who cares to go beyond a superficial appropriation of the subject: theory of linear functionals under different conditions (in X*, X**, X***, where Folland explains well the various topologies: norm, weak, weak* etc.) and the Baire category theorem along with its consequences such as the open mapping theorem, the closed graph theorem and the principle of uniform boundedness. The sections of this chapter are outfitted with plenty of good problems; for instance, the reader is asked to show that the set of nowhere differentiable functions is residual in C([0,1]). It proves frustrating, however, to dip into topological vector spaces without any significant applications exhibited. Chapter six on the basic theory of L^p spaces and duality or interpolation therein, really just a coda to chapter five, does offer some satisfying exercises; for instance, to prove necessary and sufficient conditions for L^p to be contained in L^q when p < q or q < p.
In general, Folland’s homework problems are for the most part routine and never go beyond a moderate level of difficulty. Almost always hints are provided if the problem is non-trivial. Often, the problems are intended to illustrate the theory through examples or counterexamples, but therefore do not call upon very extensive resources. It should be enough to flip through the previous section to find the relevant theorems that apply under the special case at hand. In particular, the numerous problems in chapter four on point-set topology are all straightforward.
What often happens, though, is that even if the problem itself is not intrinsically hard one can get stuck on trying a method of proof that will not work. For instance, to show that a linear operator T is closed one does not have to prove directly that for every xₙ → x, Txₙ → Tx; rather, that if xₙ → x and Txₙ → y, then y = Tx (not the same thing!). This reviewer remembers, in contrast, that in Royden’s standard beginning graduate-level textbook on real analysis (reviewed by us here), although it has a fair amount of overlap in the problems with Folland, often includes at least a few substantially challenging exercises in each chapter. A quick comparison of Folland with Royden, for the record: as to pedagogical style, Folland starts with measure and integral over arbitrary measure spaces rather than use R as a jumping-off point, as does Royden (which necessitates repetition in the later chapters on general measure spaces). Second, Folland seems to address himself to a reader who can be expected to want a more advanced point of view, and to be prepared for it.
But the choice between Folland and Royden isn’t obvious. Royden might be more thorough while Folland may give a little more advanced point of view on what he does cover. On the other hand, Folland’s homework exercises are by and large too easy, especially if one avails oneself of the hints. Suggested resolution: go through Royden first to learn the basics of the subject, then follow with Folland sometime later to refine one’s understanding (this way one can skip the standard exercises one has already done and focus on the interesting ones, of which there are many particularly in chapter five onwards; for instance, to derive Bochner’s integral for vector-valued functions and applications of Baire category including the principle of condensation of singularities).
Let us comment now on the theory of Radon measures in chapter seven. The introductory paragraph illustrates once again how Folland likes to provide helpful perspective on what he is going to do. The rather technical definition of a Radon measure shows how, back in the early decades of the twentieth century during the initial flourishing of the discipline, one has already worked all the fine functional-analytic details, issuing in a theory impressive for its scope and consequence. For instance, the inner and outer regularity properties of measures are surveyed under different conditions (σ-finiteness or σ-compactness or second countability etc.), the Riesz representation theorem and Lusin’s theorem are proved, and a good duality theory exists as well. Also, it deserves notice that the mere realization that locally compact Hausdorff spaces are about as general as one can get and still have an interesting theory of integration is itself significant. The whole theory in this chapter is a nice application of all the concepts in measure theory, point-set topology and functional analysis patiently built up in the preceding chapters, indicative of the depth of the original ideas.
This review covers the text only through chapter seven, not counting the applications in chapters eight through eleven. Arguably, it may be better to learn Fourier analysis, distribution theory and probability theory elsewhere. The main drawback of stopping here is that one has yet to get to any deep employment of all the ideas one has seen so painstakingly developed. But an education is not a matter of a day and will go on for the rest of one’s life, so we hope eventually to report on that.
In closing let us remark on the pleasure of doing real analysis in contrast with complex analysis. In the absence of the quite strong condition of holomorphicity to rely on, its results may be less elegant or ‘pretty’, but for that, also sturdier or manlier. Thus, get down to the grindstone with Folland if one ever wishes to know real analysis in order to prove something original in mathematical physics (not that is, after the manner of the field as we have it today, where heuristics and conjectures suffice, but in the old-school style where one was supposed to justify one’s contentions)!
Read
October 13, 2020I am not very analytically minded. I much prefer geometry or topological arguments or even algebraic arguments. However, I did gain much appreciation for measure theory on a second read through of the relevant sections. Moreover, the functional analysis parts of the text such as the discussion of L^p spaces were well-written enough to motivate my interest.
December 30, 2016
The story here is far uglier than its younger and more beautiful imaginary twin, which is more real than you might imagine.
June 24, 2024
To quote my analysis prof. “Folland covers all the right topics” but many times the writing doesn’t do those topics justice. At times it can shine in its clarity, the chapter on measure theory is largely well written, and at other times is confusing and vague.
January 8, 2017
I didn't know textbooks were also on this site.
Folland's Real analysis is my favorite treatment of introductory real analysis on a graduate level. It's dense, but his expository style is wonderful and in addition to the core his surveys in lesser taught areas are wonderful.
Folland's Real analysis is my favorite treatment of introductory real analysis on a graduate level. It's dense, but his expository style is wonderful and in addition to the core his surveys in lesser taught areas are wonderful.
May 13, 2016
Read the measure/integration part. Not much motivation or intuition, but good job with the choice of axiomatic development (very abstract from the outset). Terrific exercises.
Want to read
April 13, 2010likely text for math 525a
September 2, 2022
finished first 3 chapters, and some topics of functional analysis
April 23, 2007
Examples are lacking, that is nonexistent. On the other hand, the book is dense and clear.
Displaying 1 - 9 of 9 reviews