Princeton Lectures in Analysis: Difference between revisions

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{{Short description|Series of four mathematics textbooks}}{{italic title}}

{{Infobox book series

| name = ''Princeton Lectures in Analysis''

| image = [[File:Princeton Lectures in Analysis covers.gif|240px]]

| image_captioncaption = The covers of the four volumes of the ''Princeton Lectures in Analysis''

| books = {{hlist|Fourier Analysis<br>|Complex Analysis<br>|Real Analysis<br>|Functional Analysis}}

| author = [[Elias M. Stein]]<br>, Rami Shakarchi

| editors =

| title_orig =

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| illustrator =

| cover_artist =

| country = [[United States]]

| language = [[English language|English]]

| genre =

| discipline = [[Mathematics]]

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}}

The '''''Princeton Lectures in Analysis''''' is a series of four [[mathematics]] textbooks, each covering a different area of [[mathematical analysis]]. They were written by [[Elias M. Stein]] and Rami Shakarchi and published by [[Princeton University Press]] between 2003 and 2011. They are, in order, ''Fourier Analysis: An Introduction''; ''Complex Analysis''; ''Real Analysis: Measure Theory, Integration, and Hilbert Spaces''; and ''FuncionalFunctional Analysis: Introduction to Further Topics in Analysis''.

Stein and Shakarchi wrote the books based on a sequence of intensive undergraduate courses Stein began teaching in the spring of 2000 at [[Princeton University]]. At the time Stein was a mathematics professor at Princeton and Shakarchi was a graduate student in mathematics. Though Shakarchi graduated in 2002, the collaboration continued until the final volume was published in 2011. The series emphasizes the unity among the branches of analysis and the applicability of analysis to other areas of mathematics.

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== History ==

[[File:Elias Stein.jpeg|thumb|Elias M. Stein]]

The first author, [[Elias M. Stein]], iswas a [[mathematician]] who has made significant research contributions to the field of [[mathematical analysis]]. Before 2000 he had authored or co-authored several influential advanced textbooks on analysis.<ref name=oconnor>{{cite web |first1=J. J. |last2last1=O'Connor |first2=E. F. |last2=Robertson |title=Elias Menachem Stein |date=Feb 2010 |publisher=[[University of St Andrews]] |url=http://www-history.mcs.st-and.ac.uk/Biographies/Stein.html |accessdate=Sep 16, 2014}}</ref>

Beginning in the spring of 2000, Stein taught a sequence of four intensive undergraduate courses in analysis at [[Princeton University]], where he was a mathematics professor. At the same time he collaborated with Rami Shakarchi, then a graduate student in Princeton's math department studying under [[Charles Fefferman]], to turn each of the courses into a textbook. Stein taught [[Fourier analysis]] in that first semester, and by the fall of 2000 the first manuscript was nearly finished. That fall Stein taught the course in [[complex analysis]] while he and Shakarchi worked on the corresponding manuscript. Paul Hagelstein, then a [[postdoctoral scholar]] in the Princeton math department, was a teaching assistant for this course. In spring 2001, when Stein moved on to the [[real analysis]] course, Hagelstein started the sequence anew, beginning with the Fourier analysis course. Hagelstein and his students used Stein and Shakarchi's drafts as texts, and they made suggestions to the authors as they prepared the manuscripts for publication.<ref name=fefferman>{{cite news |first1=Charles |last1=Fefferman |authorlink1=Charles Fefferman |first2=Robert |last2=Fefferman |authorlink2=Robert Fefferman |first3=Paul |last3=Hagelstein |first4=Nataša |last4=Pavlović |first5=Lillian |last5=Pierce|author5-link=Lillian Pierce |title=Princeton Lectures in Analysis by Elias M. Stein and Rami Shakarchi—a book review |journal=Notices of the [[American Mathematical Society|AMS]] |volume=59 |number=5 | date=May 2012 |pages=641–47 |url=httphttps://www.ams.org/notices/201205/rtx120500641p.pdf |accessdate=Sep 16, 2014}}</ref> The project received financial support from Princeton University and from the [[National Science Foundation]].<ref>Page ix of all four Stein & Shakarchi volumes.</ref>

Shakarchi earned his Ph.D. from Princeton in 2002<ref name=duren>{{cite news |first=Peter |last=Duran |authorlink=Peter Duren |title=Princeton Lectures in Analysis. By Elias M. Stein and Rami Shakarchi |journal=[[American Mathematical Monthly]] |volume=115 |number=9 | date=Nov 2008 |pages=863–66}}</ref> and moved to [[London]] to work in finance. Nonetheless he continued working on the books, even as his employer, [[Lehman Brothers]], [[Bankruptcy of Lehman Brothers|collapsed]] in 2008.<ref name=fefferman/> The first two volumes were published in 2003. The third followed in 2005, and the fourth in 2011. [[Princeton University Press]] published all four.<ref name=ss1/><ref name=ss2/><ref name=ss3/><ref name=ss4/>

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The volumes are split into seven to ten chapters each. Each chapter begins with an epigraph providing context for the material and ends with a list of challenges for the reader, split into Exercises, which range in difficulty, and more difficult Problems. Throughout the authors emphasize the unity among the branches of analysis, often referencing one branch within another branch's book. They also provide applications of the theory to other fields of mathematics, particularly [[partial differential equation]]s and [[number theory]].<ref name=fefferman/><ref name=duren/>

''Fourier Analysis'' covers the [[Discrete Fourier transform|discrete]], [[Continuous Fourier transform|continuous]], and [[FiniteDiscrete Fourier transform|finite]] [[Fourier transform]]s and their properties, including inversion. It also presents applications to partial differential equations, [[Dirichlet's theorem on arithmetic progressions]], and other topics.<ref name=ss1>Stein & Shakarchi, ''Fourier Analysis''.</ref> Because [[Lebesgue integration]] is not introduced until the third book, the authors use [[Riemann integration]] in this volume.<ref name=duren/> They begin with Fourier analysis because of its central role within the historical development and contemporary practice of analysis.<ref name=gouvea/>

''Complex Analysis'' treats the standard topics of a course in complex variables as well as several applications to other areas of mathematics.<ref name=fefferman/><ref name=shiu/> The chapters cover the [[complex plane]], [[Cauchy's integral theorem]], [[meromorphic function]]s, connections to Fourier analysis, [[entire function]]s, the [[gamma function]], the [[Riemann zeta function]], [[conformal map]]s, [[elliptic function]]s, and [[theta function]]s.<ref name=ss2>Stein & Shakarchi, ''Complex Analysis''.</ref>

''Real Analysis'' begins with [[measure theory]], Lebesgue integration, and [[differentiation (mathematics)|differentiation]] in [[Euclidean space]]. It then covers [[Hilbert space]]s before returning to measure and integration in the context of [[Measure space|abstract measure spaces]]. It concludes with a chapter on [[Hausdorff measure]] and [[fractal]]s.<ref name=ss3>Stein & Shakarchi, ''Real Analysis''.</ref>

''Functional Analysis'' has chapters on several advanced topics in analysis: [[Lp space|L^''p'' spaces]], [[Distribution (mathematics)|distributions]], the [[Baire category theorem]], [[probability theory]] including [[Brownian motion]], [[several complex variables]], and [[oscillatory integral]]s.<ref name=ss4>Stein & Shakarchi, ''Functional Analysis''.</ref>

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== Reception ==

The books "received rave reviews indicating they are all outstanding works written with remarkable clarity and care."<ref name=oconnor/> Reviews praised the exposition,<ref name=fefferman/><ref name=duren/><ref name=ziemer/> identified the books as accessible and informative for advanced undergraduates or graduate math students,<ref name=fefferman/><ref name=duren/><ref name=gouvea>{{cite web |first=Fernando Q. |last=Gouvêa|authorlink= Fernando Q. Gouvêa |url=http://www.maa.org/publications/maa-reviews/fourier-analysis-an-introduction |title=Fourier Analysis: An Introduction |publisher=[[Mathematical Association of America]] |date=Apr 1, 2003 |accessdate=Sep 16, 2014}}</ref><ref name=shiu>{{cite news |first=P. |last=Shiu |title=Complex Analysis, by Elias M. Stein and Rami Shakarchi |journal=The Mathematical Gazette |volume=88 |number=512 | date=Jul 2004 |pages=369–70}}</ref> and predicted they would grow in influence as they became standard references for graduate courses.<ref name=fefferman/><ref name=duren/><ref name=schilling>{{cite news |first=René L. |last=Schilling |title=Real Analysis: Measure Theory, Integration and Hilbert Spaces, by Elias M. Stein and Rami Shakarchi |journal=The Mathematical Gazette |volume=91 |number=520 | date=Mar 2007 |page=172}}</ref> William Ziemer wrote that the third book omitted material he expected to see in an introductory graduate text but nonetheless recommended it as a reference.<ref name=ziemer>{{cite news |first=William P. |last=Ziemer |title=Real Analysis: Measure Theory, Integration and Hilbert Spaces. By E. Stein and M. Shakarchi |journal=SIAM Review |volume=48 |number=2 | date=Jun 2006 |pages=435–36}}</ref>

[[Peter Duren]] compared Stein and Shakarchi's attempt at a unified treatment favorably with [[Walter Rudin]]'s textbook ''Real and Complex Analysis'', which Duren calls too terse. On the other hand, Duren noted that this sometimes comes at the expense of topics that reside naturally within only one branch. He mentioned in particular geometric aspects of complex analysis covered in [[Lars Ahlfors]]'s textbook but noted that Stein and Shakarchi also treat some topics Ahlfors skips.<ref name=duren/>

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* {{cite book |first1=Elias M. |last1=Stein |first2=Rami |last2=Shakarchi |title=Complex Analysis |publisher=Princeton University Press |year=2003 |isbn=0691113858}}

* {{cite book |first1=Elias M. |last1=Stein |first2=Rami |last2=Shakarchi |title=Real Analysis: Measure Theory, Integration, and Hilbert Spaces |publisher=Princeton University Press |year=2005 |isbn=0691113866}}

* {{cite book |first1=Elias M. |last1=Stein |first2=Rami |last2=Shakarchi |title=FuncionalFunctional Analysis: Introduction to Further Topics in Analysis |publisher=Princeton University Press |year=2011 |isbn=9780691113876}}

== References ==

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[[Category:Series of mathematics books]]

[[Category:Princeton University Press books]]

[[Category:2003 non-fiction books]]

[[Category:2005 non-fiction books]]

[[Category:2011 non-fiction books]]

[[Category:Mathematics textbooks]]

[[Category:Books of lectures]]