9
May 15

V The Mathemagician’s Recommended Reading List : The Best Cheap Math Textbooks 

    We live in a transitional age where most students still derive a significant portion of their training material from a paperbound textbook.

  Despite a likely future where such textbooks will eventually vanish into history along with typewriters, horse and carriages, cocaine-laced colas, unions and the Middle Class, the  truth is that many of us-and not just us old-timers who rem-ember playing Donkey Kong in an arcade as a kid-like the feeling of a stable, physical book in our hands. This future mass extinction of the paper-bound textbook will be hugely exacerbated by the government funded textbook cartels who make certain that not only are there required text-books that cost as much as a monthly triple mortgage payment, but ensure that there’s new editions of the book every 2-3 years that differ minimally in content from their predecessors, but nearly double in cover price. (What number edition is your current calculus textbook?Bet it’s in the high single digits if not double digits.) 

It’s really amazing this textbook cabal system has been in place for a half a century-it’s been one of the root causes in the skyrocketing cost and inaccessibility of higher education in America.

What can I do about this?

Directly,not much.

Indirectly-a whole hell of a lot. 

One of the main purposes of this website is to catalog and review currently available low-cost alternatives.

One way is the use of free lecture notes and online text-books. However, I said in the introduction to the Lecture Notes Library, as rich, diverse  and useful as they are, their lack of careful proofreading  & inherently transient nature creates difficulties in using them as your sole source of education, particularly for beginners.

 Which is why I’m in the process of writing the definitive, comprehensive  e-book guide to the currently available textbooks for 30 dollars and under each, The Debtor’s Doctorate: The Mathemagican’s Guide To Affordable Mathematics Textbooks From Secondary School to PhD Level.

   The following reviews were culled from this forthcoming book-which is why you should fully expect this page to be one of the more transitory parts of the site, undergoing fairly regular revision. They will not only provide a free partial guide to the literature, but a sampling of the larger tome upon which you can decide whether or not to buy it for your further edification.  I hope the book will be completed and posted for sale in both an inexpensive paperback and e-book version by Spring 2018. 

Moreover, it's the first of a projected trilogy:Clever Cramming: The Mathemagician's Guide To The Best Mathematics Study Guides and The Forgotten Library:The Mathemagician's Guide To The Best Out Of Print Mathematics Textbooks will both be completed and available by Christmas  2019 (I hope!).

  The guide that follows are of the books I deemed, of the many I’ve read and owned, the best written and most useful of the available ones in this price range.

It goes without saying books I haven’t read myself aren’t on the list (duh). I decided not to try and list the books in order of my personal favoritism-although I reserve the right to do this at some future date if there’s enough demand for it.

 That being said, all the titles are accompanied by my review and ratings by  the same academic difficulty rating system I used for the lecture notes. My hope is to provide autodidacts looking to purchase some of the low-cost books available now a list of the books I think no library should be without.

Sadly,with a few exceptions,the highest level the books that currently exist go to is first year graduate level ( rating R on our rating system).More advanced subjects, such as operator theory and advanced algebraic geometry,don't have inexpensive hard copy books yet. Fortunately, there is a legion of online lecture notes that can be found for these subjects in the online lecture notes section.   

You'll notice there are some commonly used cheap sources that aren't present on the list,primarily study guides like the Schaum's Outlines. I decided to focus on actual textbooks in both this list and its source book because there's relatively few of them.

To me, these books-and study guides in general-require a separate treatment, which I'll provide in the second book I'm writing. When that's complete, I'll add a corresponding page to the website.   

 

So without further ado, here’s some of the crème de la creme of rustic mathematics texts.

  • Algebra by I. M.Gelfand This is the first in a brilliant sequence of precalculus textbooks planned and co-written by Gelfand to train high school students for high level university science and mathematics programs, first in the Soviet Union and later in the United States. There’s really no point in writing a list like this that begins at this level if you’re not going to include these gems. Of the tons  and tons of high school algebra textbooks out there, both new and old, this is the only one I can honestly say deserves to be recommended as a classic to everyone and their students. To me, the mark of a truly classic educational work is that regardless of level,it has something to teach everyone who reads it, from student to professional. Gelfand’s text certainly fits this criteria. A lot of books claim to begin from scratch, but Gelfand’s truly does-with basic arithmetic, done from a sophisticated viewpoint. He doesn’t build the number systems or introduce any abstract algebra, but he does ask the careful questions bright children might ask. And that’s really where all mathematical understanding really begins, isn’t it?

  •    Geometry is related to algebra and the author doesn’t hesitate to give careful arguments and demonstrations why this is so. The basics are covered-square roots, completing the square, etc.Polynomial interpolation, commonly called “long division” by high schoolers, is given probably the clearest and most careful treatment at this level I’ve ever seen.

  • There are also many basic concepts we usually take for granted in a high school student that may or may not be covered, such as arithmetic and geometric progressions-including an important application students in American schools never see anymore, the modeling of the notes of a well tempered piano by a geometric progression of frequencies. The book finishes with a beautiful flourish of basic inequalities which all of us wish we’d learned at  that level.

  •   All this beautiful stuff is presented literarily and in a crystal clear manner by Gelfand’s masterful prose with tons of simple but informative examples. This is one of the books we all wish we’d been given in high school-a book that not only informs, but inspires. The book is truly inspiring, making one think and see basic mathematics as magical and not torturous. A true classic and one no library-either for teachers,students or professionals-should be without.

    I’m estatic it’s still available so cheaply.

  • Trigonometry Gelfand

  • Trigonometry by I.M. Gelfand and Mark Saul The second book in the  aforementioned sequence of precalculus textbooks builds beautifully on the first. 

  •  

    Interestingly, you’d think the next volume in the series would cover basic Euclidean geometry. Gelfand did in fact co-write such a textbook with T.Alekseyevskaya-but it has never been available in English to students outside the Gelfand Correspondence School. It should be noted that in 2014,the English version was submitted for publication. Sadly,it has not yet appeared.)  

  • The theme of this book is to use the trigonometric functions to establish the careful connection between basic algebra and basic geometry that results in the creation of analytic geometry.  Again, all the basics are developed-the sine, cosine and tangent functions, the Laws of Sine and Cosine, the Pythagorean  Theorem and more. 

  • There are tons of worked examples and challenging exercises.  But the central idea throughout is that trigonometry is the study of  periodic functions and these periodic functions are critical throughout mathematics, particularly calculus. The book is specifically geared to prepare students for calculus. One of the main problems most of today’s students in basic calculus have-among many-is weakness in trigonometry. A careful study of this book would help such students immensely. Another classic where the authors treat beginners from the standpoint of mature mathematics while still remaining at the appropriate shallow end of the pool. Most highly recommended.

  • Functions And Graphs Gelfand

  • Functions and Graphs by I.M. Gel'fand ,E.G. Glagoleva and E.E. Shnol : (G) The third book in this wonderful series by Gel’fand and his co-authors is exactly what the title says it is- a mathematically sound tome for beginners on the myriad of functions and their graphs in the plane and how the latter informs us greatly about the behavior and properties of the former. It’s wonderfully pictorial and intuitive, with graphs on every page and every conceivable function one would find in pre-calculus courses. The main emphasis of the book is on the technique of “shifting” graphs i.e. translation through the  coordinate system of the Cartesian plane. This is an absolutely critical basic skill for students and it’s surprising how many calculus and geometry students are weak in it.  Together with the aforementioned book on trigonometry, a good high school geometry course textbook, such as Kiselev’s texts to be discussed in the next section-I can’t think of better preparation for a strong calculus course. Another jewel by Gelfand and his coauthors for beginners.

  • Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry by George F. Simmons:    This was the last of the amazing textbooks by the late Simmons, which include such classics as An Introduction To Topology and Modern Analysis and Differential Equations with Applications and Historical Notes. Each of these books is marked by Simmons’ unique combination of detailed exposition, lively
    prose and conceptual clarity. This-his last and most elementary textbook- is no exception.

  • The title is very indicative of the contents-the purpose of the book is to present clearly and carefully the critical
    minimum of algebra, geometry and trigonometry needed to effectively and carefully learn calculus.The contents are therefore highly selective and focused-nothing is developed which isn't  needed to understand either calculus itself or the analytic geometry which is inseparable from it and it's applications. By necessity, the book is less detailed then the texts by Gefland above. But for students and teachers that are interested in a comprehensive but focused introduction to the fundamentals needed before embarking on a serious calculus course such as Silverman's below, this is as good as you're going to find. The book has 3 sections: algebra, geometry and trigonometry-and all 3 are covered with both care and clarity. There are many, many pictures as well as the author's usual dry wit and masterful  organization. The last chapter on trigonometry,in particular, is probably the best short introduction to the subject that currently exists.

  •  The great shame of this book is that it's the only one by Simmons that's currently available in our price range. We should all push for the remarkable textbooks of this fine author and mathematician to be reissued in cheap editions for future generations. In the meantime, beginning students and teachers have this jewel to treasure.

  •    

  • Modern Calculus and Analytic Geometry  Yes, that Silverman, the one behind many excellent translations of classic Russian textbooks that Dover has republished so many of-such as the 3 volume treatise on complex analysis by Markushevich. Apparently this was one of the first books he completed after finishing his PhD at Harvard in the 1960's.

  •    Back then, during the Golden Age of American Education, this could pass for an honors calculus text at an average university. No longer. Now, it would be a truly superior student indeed who could handle the more theoretical parts of the book-which includes precise definitions of ordered pairs, relations and functions, a full development of the real numbers based entirely on decimal expansions rather than Dedekind cuts or Cauchy sequences. This last part in particular is fascinating since it’s a method of construction of  R that mathematicians usually avoid like live HIV virus because it’s by far, the most complicated and tedious method of constructing the reals from the rationals.  But the advantage of this method is that conceptually, it is the simplest construction. This demonstrates the underlying philosophy of the book: Maintain the intuitive, physical/geometric aspects of calculus, but prove everything in the simplest manner possible.

  •   The result is a completely rigorous presentation of both single variable and classical multivariable calculus in modern language (i.e. done with rigorous set-theoretic constructions and linear algebra but no manifolds or differential form theory). The book also has a lot of nice examples, which is unusual in calculus books this old. This is probably my favorite of all the Dover calculus books and I’m seriously considering trying to build an honors calculus course around it.

  • The Calculus Lifesaver: All the Tools You Need to Excel at Calculus  A new classic. This is the book all of us wish we’d had when we first started learning calculus. In softcover, it’s cheap enough to make the list.  Adrian Banner developed this book out of an intensive tutorial session for freshmen and high school students he taught for several years-and the experience shows. It is rigorous without being abstract, has tons of pictures to go with the beautiful, well-thought out explanations and proofs and almost 500 step-by-step solved problems. Banner’s explanations should be studied by any budding teacher or professor as models of clarity and depth, teaching not only the concepts of calculus but problem solving techniques and strategies. All this and for free, you get the legendary video series of Banner’s lectures as well as additional video lessons at http://press.princeton.edu/video/banner/.  This is a must-have, not only for all students taking calculus for the first time, but all instructors teaching it. The one drawback it has is that it only covers single-variable calculus. Here’s hoping sections on multivariable calculus and basic linear algebra are in the offering from Banner in either a second edition or a sequel.

  •   

  • Calculus for the Ambitious by Tom Korner I did a double take when I first learned this book was coming out. Korner is not only a first-rate mathematician, he may be the very best mathematics textbook author going these days. From his wonderful “shop-window” text on Fourier analysis to his tour de force on intermediate undergraduate real analysis to his incredible recent text on linear algebra for both mathematicians and physicists, he has authored some of the deepest and most vivaciously written texts for serious students that are currently available.

  • This is not a textbook on calculus in the formal sense.It’s an informal supplementary text for such a course and for young mathematics students just beginning their studies in calculus and have no clue what it’s about. And just like in all of Korner’s other textbooks, it’s incredibly well written, masterly and has remarkable insight into its chosen subject matter like no other book in its genre.  In just 180 (!) pages, Korner gives virtually complete coverage of the basic ideas of calculus-from basic ideas of derivative, limit, continuity and integral to the elements of single and multivariable calculus through the basic questions surrounding the real numbers and the imprecision of “intuitive” calculus through the basic ideas of the construction of the real numbers and the beginnings of rigorous analysis.

  •    There are no formal proofs, but on every page, there’s a plethora of ideas and razor sharp clarity in the exposition of those ideas. So many of us learn and/or teach calculus, but so few of us can really present a truly deep conceptual understanding of the immensely subtle and beautiful ideas of calculus in a first course to beginners. Korner doesn’t just do this, he provides a complete and incredibly deep deconstruction of the basic ideas of both calculus and analysis  in a literary manner that will provide remarkable insight into this central subject for all readers-from high school to seasoned teacher of calculus or real analysis. A remarkable course in calculus could be taught from it in conjunction with a standard text. It’s the kind of book that regardless of how many times you pick it up, you’ll find a new comment or insight you didn’t see before. I’m so happy this amazing book is available for a mere 28 dollars. Please go get a copy, regardless of your level of calculus knowledge. It will become one of your most used and treasured texts in your library, trust me.

  • Finite Dimensional Vector Spaces by Paul Halmos (PG/PG-13)If you’ve never heard of the late Paul Halmos, then you’ve been sadly deprived in your education. Halmos was a Hungarian born, American raised mathematician who worked at several major universities in the US,  most notably the University of Chicago.  He was probably best known for his remarkable autobiography, I Want To Be A Mathematician- which was one of my major inspirations for switching from chemistry to mathematics. I still reread it from time to time for not only inspiration, but Halmos’ wonderful commentary on mathematics which is so eminently quotable. I heartily recommend anyone interested in mathematics and the practice of mathematics to beg, borrow or steal a copy.

  •  In addition, he was known for his advanced textbooks on a host of topics for which there were no accessible treatments before his textbooks. Most of his textbooks have recently been reissued by  Martino Fine Books in very inexpensive paperbacks-which is great news for all of us. This was probably his most influential one.

  •    FDVS was a historically critical book-it was the first real textbook on linear algebra. Until Halmos wrote it, the teaching of linear algebra as a separate subject to undergraduates was not something that was common in university curricula-as strange as that seems to students and professors today. Matrices, vector spaces and linear transformations were topics that were either taught in detail as part of abstract algebra courses or on a need to know basis in graduate courses in functional analysis. By the early 1960’s, the need to introduce linear algebra much earlier and in much greater detail in a students’ training as a foundation for a myriad of subjects was generally accepted. Halmos’ book went a long way towards making this revision practical.

  •    That being said, those who expect a standard linear algebra text are going to be taken aback when they open it. Halmos’ inspiration for the book was the first part of John Von Neumann’s lectures at Princeton in the 1940’s on operator theory, which were pitched at graduate students and professionals. What Halmos wanted to produce was a finite-dimensional version of the vector space concepts in those notes for students preparing to study functional analysis. As a result, the book is pitched at a much higher level than one expects in a modern linear algebra course. The emphasis is on linear transformations and dual spaces-matrices don’t even appear until halfway through the book. The book also has many sidebars into modern analysis in finite dimensions, such as bounded normed spaces and an introduction to Hilbert space. While these sidebars are important and fascinating, it’s kind of hard to justify such topics in a beginning linear algebra course these days.

  •    It’s also quite a bit terser then books on linear algebra tend to be-almost half the results are in the exercises. The author was a firm believer that mathematics has to be learned with pen and paper in hand-and it’s  hard to argue with this sentiment. As a result, even talented beginning students with little rigorous mathematics background may find it overwhelming.

  •    That being said-Halmos is one of the clearest, most insightful mathematics textbook authors that has ever put pen to paper. The book has many examples-most drawn from classical geometry, where the connection between the 2 subjects is one of the book’s focuses. His book is a model of clarity and students that work through it will come away with a wonderful understanding of the subject-and a better understanding of abstract linear mappings then most standard books give. I’d have a lot of trouble recommending it to students just beginning to learn these concepts, even really strong students. But it certainly would make a terrific first course for honors students or a second course for math majors-and at this price, why in the name of the ghost of Irving Kaplansky wouldn’t you want a copy?   Highly recommended for strong and advanced students only.

  •   

  • Vector Calculus by Peter Baxandall and Hans Leibeck (PG)       I stumbled across this terrific and very underrated book while searching for a modern treatment of functions of several variables that could be used by bright undergraduates without the use of manifolds or differential forms. This is one of the best available. 

  •    The overwhelming majority of such books are basically plug and chug books that may as well have been written in the 19th century and avoid any hard theory like Lyme Disease. Most of them are afraid to discuss linear transformations and vector spaces, for God’s sake. Such books are obviously mostly written for engineering and non-mathematics students, assuming that honors mathematics and physics majors would opt into courses and texts based in differentiable manifolds. Like I said above, this is a highly questionable assumption to say the least and even if it were true, a careful study of vector valued functions in R2   and R first might allow students to transition directly to modern differential topology and geometry without the “forms for dummies” approach. In any event, this wonderful textbook is rooted in the UK university system and therefore begins at a higher level them most American texts. 

  •     The authors presume the students have had courses in linear algebra and strong one variable calculus using ε-δ  limit definitions. This makes a world of difference as it allows them to present the elements of several variable calculus as the study of certain linear transformations (the general derivative, the differential)  between subspaces of R 

  •     The book has a “spiral ascent” structure-it begins with the simplest kinds of functions of several variables, namely  the real valued maps of Rand proceeds through vector valued maps in R2 then maps from  R2  to R3 , etc., culminating with a careful study of vector valued functions,
    derivatives and differentials, and line and surface integrals in general Rn  .  
    This  way  the presentation
    begins in the simplest manner and gradually achieves full 
    generality. The standard concepts-such as chain rule, the inverse and implicit function theorems and multiple integrals-are presented several times at different levels of generality.

  •    The language of linear algebra is used freely and without reservation, careful definitions are given & the presentation is still  extremely visual as each concept is given with several graphs. Even better, the presentation is  example driven-there are literally hundreds of examples throughout-from both mathematics and classical physics. And it’s all topped off with lots of equally terrific exercises, none too hard.

  •   This is the kind of book mathematicians and physicists wish they’d had when they first learned vector calculus. I remember reading it and thinking how much easier Barrett O’Neill’s differential geometry book or Spivak's Calculus on Manifolds would have been if I’d mastered Baxandall and Liebeck first. An absolute must for any student trying to master multivariable calculus.It’ll also make very helpful collateral or prior reading for any student about to take a course in differentiable manifolds or
    differential geometry. The 
    highest possible recommendation!

  • Introduction to Analysis by Arthur Mattuck (PG)     I was absolutely floored to learn this incredible book was reissued in a fantastically cheap paperback after becoming exceedingly scarce-and correspondingly expensive-in it’s original hardcover edition. Mattuck’s reissuing of the book through Createspace in a 13 USD price edition as the standard text for the 18.100A course at MIT, which he has taught there for many years, demonstrates this legendary teacher’s commitment to assisting students learn analysis who do not have the strong background and/or talent of students who can directly enter an abstract analysis course based on metric spaces.

  •    The author’s “mission statement” of the text in the Preface is quite informative and should be read by all students and teachers of mathematics. Basically,the need for such a text became clear from Mattuck’s many years of teaching analysis to students who struggled in the typical Rudin-based analysis course. (And these were MIT students, so the usual snarky response of mathematics honors students that they just weren’t bright enough to learn analysis becomes manifestly absurd in this case…..) A particular incident the author recalls brings into sharp focus his perception of the problem to such students:

  •  Years later, a senior physics undergraduate sat down before me and sighed. “Well,this is the fourth time I’m dropping analysis. Each time I get a  little further into the course, but the open sets always win out
    in the end. Isn’t it possible to teach it so guys like me could understand it? We understand derivations, but they give us proofs instead. Inequalities are OK, as long as they look like equations, but this analysis doesn’t look like the math we know — it’s all in English instead of symbols. And as far as any of us can tell, the only thing any theorem is good for is proving the next theorem.” 
       

  •    The problem here is not lack of mathematical talent, but rather a sharp difference in the  perception of mathematics between the usual mathematics majors compared with bright students of an non-pure mathematics background-such as physics majors or students coming late to a mathematics major from other disciplines-who want wish to understand the theory of calculus. After excelling at computations, students of the latter category have at best a middling understanding of calculus and the real numbers. They are utterly confused by the level of abstraction when ε-δ arguments on the real line-which really underlies all the calculus they’re used to-is bypassed entirely for the machinery of metric spaces. Also, this may be the first mathematics course for which they are required to do serious proofs-and it may be the last course they really need where they have to. By contrast, mathematics majors usually learn this material in honors calculus concurrently with many proof techniques and applications in not only their calculus course,but other courses.  How the book differs from more sophisticated treatments is best described by the author, so we quote again:

  •     The book is basically one-variable analysis. The emphasis throughout is not on the algebraic or topological aspects of analysis, but on estimation and approximation: how analysis replaces the equalities of calculus with inequalities, certainty with uncertainty. This represents for students a step up in maturity.To help, arguments use as little English as possible, and are formulated to look like successions of equations or inequalities: derivations, in other words Basic one-variable calculus is used freely from the beginning as a source of examples, so students can see how the ideas are used. The real numbers are discussed briefly in the first chapter, with most of the emphasis on the completeness
    property.The aim is to get to interesting things as quickly as possible. Several appendices present extended applications.  Point-set topology, the pons asinorum of analysis courses, has been banished to near the end, and presented in abbreviated form, just before it is needed in the study of integrals depending on a  parameter. By then, students can understand the arguments, and even enjoy them as something new-looking.
  • To these ends,the author takes great pains to give proofs in great detail, all done by specific calculational methods at first, which slowly give way to more general arguments as theorems are established
    throughout the book. The Completeness property is given in terms of Cauchy convergence. The author writes beautifully and clearly, with many deep insights that are usually omitted as obvious in
    not only analysis courses, but calculus courses. For example, he goes into some detail on why subtracting inequalities is illegal.  He also can be wryly amusing at times. There are tons of excellent problems
    all with complete solutions, which will make the book incredibly useful for self-study. The book, to me, has 2 flaws that would be rather easy to fix in a second edition: Firstly, there’s no clear development of the number systems from the natural numbers through the reals, even an axiomatic one. There’s a scattershot discussion of the number systems in appendix A, but I think the beginner will find it more
    confusing then helpful. The author  understandably decided to omit it in favor of getting the students’ hands dirty working concrete examples with his accompanying commentary. But I think a detailed appendix at least outlining an axiomatic development is in order in such a book since understanding the real numbers is so critical to understanding the underlying material. Also, I wish he’d included a guide for further study to direct self-study students where to go next. These are really minor
    complaints, though. Mattuck has written an outstanding textbook that all students of mathematics regardless of level can learn from. It’s  an absolute gift that the book is now available so cheaply to
    beginning students of analysis and there’s no reason not to have a copy now. Run and order a copy. You’ll thank me later, trust me.

  • Analysis in Euclidean Space by Kenneth Hoffman I had the pleasure of reviewing the Dover edition of  this book
    several years ago for the Mathematical Association of America’s 
    online reviews page. My opinion of the book
    hasn’t changed much, so this 
    will largely be a truncated version of that review.

  •    The late Hoffman was truly one of the giants of modern analysis from the mid-20th century onward and you’d certainly think an undergraduate analysis text authored by him would be more widely used. It developed out of Hoffman’s lectures on undergraduate real analysis at MIT beginning in the late 1960s. The author had a reputation among the students at MIT as passionate teacher nd communicator of mathematics. Judging from this
    wonderful and unusual text, 
    that reputation was well deserved.

  •   The book is clearly intended as a text for the first real analysis course for serious students with solid training
    in geometry, calculus, and linear algebra-the book is at 
    approximately the same difficulty level as "baby Rudin". However, there are two  major differences between Hoffman’s text and the standard books at this level. Firstly, the emphasis is much more on why then on what. Hoffman expends a great deal of time and effort explaining what he calls the “4 Cs” of basic real analysis: convergence, compactness, continuity and connectedness. Chapter 1 gives a detailed review of the vector space properties
    of Rn .

  •    This is highlights the second and probably most important difference in the text as opposed to the usual present-ations at this level: the development is based entirely on the fact that Euclidean spaces (including R and C, of course) are normed linear spaces. Metric spaces and topology are not discussed in depth; metric spaces are defined only on page 260(!) at the end of the discussion of general normed spaces and the word topology is only mentioned once during the discussion of sequential compactness and relative openness of subsets of Rn. There are 2 big advantages to this approach. The first is that
    it’s 
    inherently very visual because then the geometry of Ris naturally tied to the analytic properties. There are many, many pictures — the geometric perspective makes this very natural and not forced as in many analysis texts. Also, not only does it allow a unified treatment of the real line and all its generalizations to higher dimensions, it also provides the natural basis for later studies of functional  analysis and harmonic analysis, which generally emphasize the properties of Hilbert and Banach spaces as normed rather then topological spaces. So a course based on this book provides a natural and straightforward foundation for such advanced courses.

  • Hoffman’s book has enormous breadth and depth-not only does it cover the usual topics, like limits of sequences and series and the derivative, the emphasis on normed spaces
    allows coverage of 2 more sophisticated topics which act as a 
    prelude to functional analysis courses: general normed spaces and the Lebesgue integral a la Daniell. The exercises and  examples in this book show this is not merely a
    compendium of facts — Hoffman wants students to 
    not
    only learn analysis, but to think about it. The book’s 
    organization clearly demonstrates the deep and original perspective of the author on the subject and his willingness to put in the effort to pass this perspective to his students. In short, this is a buried treasure unearthed by Dover and now there’s no good reason not to have a copy.Highest
    possible recommendation.

  • General Topology by Steven Willard (PG-13)  I was absolutely stunned when I found out Dover had reissued this book a few years ago. I first discovered this classic reference while taking point set topology as an
    undergraduate and it still amazes me. It’s probably the single most 
    complete reference/textbook on the subject that’s ever been written. 

  • Willard’s treatment beautifully encompasses
    virtually the whole of point set theory;he distinguishes two 
    broad areas of topology: "continuous topology,"
    represented by sections
    on convergence, compactness, metrization and complete metric spaces, uniform spaces, and function spaces; and the beginnings of "geometric topology," covered by nine sections on connectivity properties, topological characterization theorems, and homotopy theory   (Homology theory isn’t covered , but this is a book on general not algebraic topology and that properly belongs in the latter. )In each section, there
    are a wealth of examples and terrific exercises and the book 
    concludes with a very informative and scholarly
    overview of the history 
    of the development of each topic in point set topology. Even if you don’t particularly care for point-set topology-as sadly a lot of research oriented students don’t and are encouraged to feel that way-there’s really no good reason not to have this wonderful book now.  

  • Before moving on, I just wanted to alert readers who may not be aware of it: Besides Willard, there are 2 other equally comprehensive textbooks on general topology, the books by Ryxzard Engelking and James Dugundiji. Engelking is nearly unobtainable in the U.S and it’s insanely expensive to order from Heldermann-Verlag. Dugundiji is long out of print. So until either becomes available in the U.S. again in inexpensive editions, Willard is really your best bet for such a text. 

  •  
  • Topological Methods in Euclidean Spaces by Gregory Naber (PG) Alright, everyone stop reading and pay attention.

  • Everyone. 

  • Ok?

  • This is the single best textbook on topology that has ever been written for undergraduates. Ever. Got that?

  • I’m dead serious. I know I’m going to get a lot of blowback and fights for this statement and it’s still possible I’ll find a better one or change my opinion in the coming years. But as it stands right now, this is the best one ever for anyone who doesn’t know anything substantial about topology and is seriously thinking about graduate school in mathematics.

  • Better then Armstrong.

  • Better then McCleary or Crossley.

  • And yes, for  undergraduates, better then James Munkres' classic-may the Gods of mathematics forgive me and may James Munkres forgive me!  

  • This is a practically forgotten textbook by a well-known expert on mathematical physics and gauge theory at Drexel University, who has since built a strong reputation in both
    research and teaching, being 
    know for such unorthodox but masterly textbooks as The Geometry of Minkowski
    Spacetime
    and the 2 volume 
    definitive treatise on the mathematics of gauge theory, Topology, Geometry And Gauge Fields. 

  • (as well as Naber’s online  lecture notes for his courses, which we’ll get to later)

  •   These texts have become standard sources for both mathematicians and physicists alike
    due to Naber’s remarkable penchant for seamlessly integrating mathematical rigor and comprehensiveness with a talent for concrete computations and examples. This book is very much in that tradition. The entire text is laid out entirely in Euclidean spaces without the abstract definition of a topological space appearing anywhere in the text. This is part of Naber’s strategy of utilizing all the familiar machinery of both calculus and geometry to describe the concepts of “nearness” (metrics) , open and closed balls, continuity, connectedness, compactness, the fundamental group and homotopy theory, simplexes and their associated techniques such as the Brower theorem and triangulations, the basics of homology theory in Euclidean spaces, including an elementary discussion of homological algebra and diagram chasing!  

  •    The amazing thing is that all this beautiful material is described very geometrically with many pictures and yet in completely modern language-and at a level any student with a standard US undergraduate mathematics major can understand. Many wonderful examples are embedded in the presentation. Naber concludes his textbook with a great list of recommendations for further reading, containing commentary on some of the subject’s true classics.  It’s hard to imagine any undergraduate who works through and masters this wonderful book having anything less then a full overview of the entire field of modern topology in Euclidean spaces when they finish and they’ll be well-equipped for graduate courses. I can’t recommend this book highly enough!

  • Counterexamples in Topology by Lynn Arthur Steen and J. Arthur Seebach Jr. This was the first of 2 famous textbooks on counterexamples reissued by Dover, the other being the one on basic analysis here. I don’t think this book is as well organized or written as Gelbaum and Olmsted’s book. That being said, just about every wacky topological space and crazy mapping on them is described in here and in detail and it even contains a rapid review of point set topology. So since Dover has it available so cheap and this material can’t really be found elsewhere without half a dozen topology textbooks on hand, why shouldn’t you get it?

  •  
  • Number Theory by George Andrews This unorthodox introduction to number theory for undergraduates uses combinatorial methods to develop the theory independently of either algebraic or analytic methods, particularly the theory of partitions. Andrews was an eminent number theorist and combinatorialist at Penn State University and he brings his expertise in both subjects to bear producing a purely combinatorial presentation of
    the properties of the  
    integers. 

  • For example, he proves Wilson’s theorem by a rather lengthy but very clever and clear graph-theoretic argument of the number of edges that can be constructed from a bisected circle with p points on the boundary. Generating functions are introduced early and used as a major calculational tool in the derivation of partition generating functions, which is used to give an introduction to additive number theory (one of my areas of research, which has expanded dramatically  since this book was written) .

  •   Surprisingly, Andrews also shows a converse relation i.e. how results in number theory can be used to solve problems in combinatorics. For example, he demonstrates how Fermat’s little theorem can be used to derive results about riffling phenomena. Wonderful and unexpected results like this are sprinkled through Andrews’ book-this plus the literate, relaxed style of his writing make this a very original classic of the subject and there’s absolutely no reason not to have it in this cheap edition.

  •  
  • Ordinary Differential Equations by Morris Tenenbaum and Harry Pollard  This is one of the all time great classics on the subject for undergraduates and it amazes me how many
    students taking differential
    equations for the first time are still completely unaware of it. 

  •   One of the marks of a true classic in the sciences is that it ages well i.e. very little of it has been invalidated by subsequent progress in its field and it can still be used with enormous profit by students and experts even if its’ generations later. Van deer Wearden’s Modern Algebra is such a text, so are John Milnor’s Topology From A Differentiable Viewpoint and Morse Theory, John Kelley’s General Topology and Ira Singer and John Thorpe’s Lecture Notes In Elementary Geometry And Topology, just to name a couple.In differential equations, my favorite is the true classic, George Simmons’ Differential Equations
    with Applications and Historical Notes.
    (In fact, the late
    Simmons was one of the great textbook authors in mathematics and he has written several textbooks which were gems for the ages, mostly in calculus and topology.)  My point is that Tenebaum/Pollard is of the few texts on DE, in an age of literally tons of contenders, that I would recommend or assign to minimally prepared undergraduates first learning the subject who are serious about it.
    As Simmons has aged well and is just as valuable as it was when it
    was published 40 years ago-so is Tenebaum/Pollard, which is a decade
    older. The authors wrote their book for a mixed audience
    of mathematics, physics and engineering students and therefore make
    no real concession for the makeup of their audience. That is to
    say, they discuss careful theory and physical applications in equal
    measure in their book and for differential equations, the
    importance of this cannot be overstated. Both aspects of the theory
    of ODE are of equal importance in understanding it. One cannot
    truly understand the solution of the one-dimensional oscillator
    without its derivation from the period of compressed elastic
    spring-mass system. Conversely, one cannot completely understand
    this derivation without understanding the purely mathematical
    properties of second order linear differential equations.
    Tenebaum/Pollard explains both aspects very clearly with lots of
    fully explained examples, developing both the students’ theoretical
    and calculational abilities-both of which are equally important n this
    subject. The one minor flaw in the book is the old-fashioned
    notation and presentation-linear algebra isn’t explicitly used and
    that makes the later chapters about systems of ODE’s a bit awkward
    and tedious. But that’s easily corrected by rewriting it in matrix
    language, a task for the instructor. A must have for either the student
    taking differential equations for the first time or the young
    professor teaching it for the first time.

  •  
  • Introduction
    to Partial Differential Equations with Applications by E. C.
    Zachmanoglou and Dale W. Thoe
    Unless one takes a purely

    applied approach and doesn’t worry
    about the underlying theory-which many books 
    written
    for physicists and engineers happily do- an undergraduate 
    presentation of PDE’s at a level requiring no
    more then calculus and/or 
    linear
    algebra is very difficult. There are several older books that 
    attempt to bridge this gap and present a fairly
    rigorous introduction to  
    PDEs
    while limiting their domains to finite dimensional Euclidean
    spaces. 
    This has the advantage
    of allowing a fairly careful approach to the 
    subject,
    sacrificing generality for specialized rigor. At the same time, 
    since the physical application of PDE’s is even
    more important then in the 
    case
    of their one variable counterparts, one does not want to slight
    the 
    reader in concrete
    examples and important applications to the physical 
    sciences.
    Because of all these considerations, I think Zachmanoglou/ Thoe 
    is one of the very best books in this genre and
    it has aged remarkably 
    well in
    nearly 40 years. The book developed from an undergraduate course 
    they taught at Purdue University for nearly a
    decade whose purpose it was 
    to
    present to undergraduates a careful and modern, but reasonably 
    elementary, presentation of all the basics and
    applications of PDE’s. The 
    assumed
    prerequisites are a year of “advanced calculus “(i.e. a first 
    semester of real analysis of one variable or a
    strong honors calculus and 
    a
    subsequent semester of a rigorous course in functions of several 
    variables in Rn) , a course in linear algebra
    and a course in ODE’s, These 
    are
    very reasonable to provide a foundation for introducing the subject
    at 
    the undergraduate
    level.  There are many pictures, many in R
    , of the geometry of the given equations and
    their solution spaces, 
    including
    a detailed discussion of the integral curves. It’s clear the 
    authors are trying to provide an in-depth
    presentation of classical PDE 
    theory
    in the plane and in “3-space” while using completely modern
    language to present it. The book neither skimps
    on theory or 
    applications-there
    is a full chapter on mathematical physics and the major 
    equations like the wave equation and there’s a
    full chapter on the Laplace

    equation and is role in the solving of linear PDE’s. Lastly, each
    chapter 
    ends with a very
    thorough set of references for each subject for further 
    reading for the serious student. This is a
    superb book on a very difficult

    subject and I can’t think of a better one for absolute beginners then
    this 
    one. Anyone who studies
    this book will be well prepared for a graduate 
    course
    on the subject (once they’ve completed the equivalent of a first

    year graduate course in analysis, of
    course).

  •  
  • Introduction
    to Graph Theory by Richard Trudeau
     This and

    Hartfield/Ringel were the textbooks for my first serious graph theory

    course, taught by the eminent graph
    theorist John W. Kennedy.(On a side notes

    Dr. Kennedy disappeared from the faculty of Queens College Of CUNY 
    several years ago and where he’s gone is one of
    the great local unsolved 
    mysteries
    of my alma mater. There’s been all manner of mad theories

    circulating about his current whereabouts: From him becoming a
    recluse 
    to work on a new
    generation of graph theoretic software for quantum 
    computers
    to working on top-secret Enigma-like codes for Homeland

    Security or the armed forces in the Afganistan war to the previous 
    theory’s Jungian shadow; that he was in fact, an
    Al-Quieda operative 
    infiltrating
    our academic system, hoping never to be discovered at such 
    an undistinguished university-and a black ops Navy
    Seal Team threw a 
    burlap sack
    over his head as he walked to his car from the mathematics 
    department, threw him in the proverbial unmarked
    black van bound for
    Gitmo and never
    to be seen alive by human eyes again. All these wild 
    theories
    aside-Professor Kennedy, if you’re reading this, please let me 
    know what the reality is if you can. I will say in
    addition-many of your
    former
    students, me included, miss you, your dry wit and your lecturing 
    brilliance. ) In any event, this book was chosen
    by him because under 
    it’s
    original title, Dots And Lines, it
    was the book he used as an undergraduate at Cambridge to learn graph 
    theory 4 decades ago. Despite the enormous
    progress and development 
    that’s
    been made in this field since its original publication, it 
    remains a surprisingly strong introduction to
    graph theory, still 
    containing
    all the essentials: vertices and edges, simple graphs, graph
    multiplicity, connectivity, planarity and
    nonplanarity of graph plane 
    embeddings,
    dual graphs.the graph theoretic version of Euler’s formula 
    and much more. There are lots of pictures and very
    good exercises for
    the beginner.
    No, it’s not very up to date and a lot of more recent 
    topics-like
    more recent coloring results-are missing.  But it’s 
    still a very good and cheap introduction.

  • General Theory of Functions and Integration by Angus Taylor  For the second time
    in this list, I’m calling for everyone’s attention. And this time
    mean it. Ready? Ok? This
    is not only the best advanced analysis 
    text
    I've 
    ever
    seen, it’s my favorite book on 

    this list. 
    If
    any young professor is thinking of writing 

    an advanced textbook for graduate students and doesn’t have a clue how
    to 
    go about it, he/she needs
    to stop everything and run out and get this 
    book.
    Studying the structure of this book will not only teach you a lot
    of 
    wonderful mathematics you
    may have forgotten (or not learned in the first 
    place!)
    , but it will certainly demonstrate how to write textbooks for 
    advanced students. It must be informative but
    not effortless, clear but

    challenging and its exercises must be difficult enough to build 
    mathematical muscle, yet not be impossible for
    any but a genius. Taylor’s 
    book
    does this better then just about any book I’ve read-I keep 
    discovering new treasures in it each time I go
    back to it. It contains 
    everything
    you ever wanted to know about not only basic measure and 
    integration theory, but point set topology on
    Euclidean spaces in both 
    metrics
    and norms, vector space theory leading to the basics of Hilbert 
    and Banach spaces,  basic inequalities in
    both integration and
    functional
    analysis, full presentations and comparisons of both the 
    measure-theory based construction of the
    Lebesgue integral as well as the 
    Daniell
    approach via simple functions and much more.Indeed, a detailed
    development of both approaches and showing
    their equivalence is one of the 
    main
    purposes of the text. Several points sum up my advocacy for this
    book 
    for all students of
    analysis. Firstly, I believe this is the perfect text

    to supplement "Papa Rudin". Not only does it cover incredibly
    important material that 
    every
    graduate student should know in a beautiful and thorough manner,
    it 
    really covers specifically
    what Rudin either skims over or goes through 
    far
    too quickly for such students to be able to master. Working 
    simultaneously through Taylor's opus
    concurrently with the Green Bible 
    will
    vastly improve both their speed and depth of comprehension of
    modern 
    analysis. Even better,
    any student who studies Taylor in the summer months 
    before
    a graduate analysis course based on Papa Rudin is going to find it 
    far easier to conquer then it would be
    otherwise. Secondly - the exercises 
    in
    Taylor are outstanding. Not only do they run the gamut
    from 
    simple proofs and
    calculations to major problem sets, how they are 
    constructed
    will be of enormous help to students. One of the things I love 
    the most about these exercises is how Taylor
    supplies the best hints I've 
    ever
    seen to substantial exercises - without giving the punchline away,
    he 
    instinctively seems to know
    exactly what piece of information to give the 
    good
    student to point her in the right direction. This is the work of a 
    gifted teacher. Thirdly, it has many excellent
    pictures and examples-which 
    is
    so unusual in books at this level. Lastly - it's in Dover,which
    means 
    there's absolutely no
    good reason not to have a copy. You can probably get 
    one
    used for under 5 dollars. Not getting such a remarkable work so
    cheap 
    is a crime. Period. Get
    yourself a copy. You'll thank me.


  • Functional Analysis by George Bachman and Lawrence Narici
    (PG-13) For over a generation, this was the introductory
    textbook on the subject for strong undergraduates and 
    self-studying graduate students-and it richly
    deserves to continue to have 
    that
    exhaulted position. Sure, there are a few textbooks suitable for
    strong 
    undergraduates now: Barbara
    McCluer’s Elementary Functional
    Analysis
    jumps immediately to mind. But
    let’s 
    face it-the vast
    majority of the standard textbooks are at the very least, 
    first year graduate student texts and are
    accessible to undergraduates only 
    at
    the very best schools. Most require some facility with measure theory
    and 
    abstract algebra. Bachman
    and Narici requires only a basic real analysis 
    course
    and a good working knowledge of linear algebra. With such a meager 
    background, you’d think they couldn’t cover much
    in such a course, but boy, 
    would
    you be wrong. This book covers just about everything you want in a 
    first course in functional analysis and it does
    it in a very gentle and 
    detailed
    way-without sacrificing rigor or challenge to the students. The 
    book covers in succession as follows: The first
    2 chapters cover fully 
    finite
    dimensional inner-product spaces up to the spectral decomposition
    of 
    a. normal linear
    transformation into a linear combination oi orthogonal 
    projections. The next. four chapters are devoted
    to the properties of metric 
    spaces
    including the category theorem for complete metric spaces.Chapter 
    seven gives a brief introduction to general
    topological spaces with special 
    reference
    to compactness and gives a proof of the Tychono? theorem on the 
    compactness of the Cartesian product of compact
    spaces. Chapter eight deals 
    with
    normed spaces and Banach spaces with numerous illustrations. Chapters
    nine and ten are devoted to
    an introductory exposition of Hilbert spaces,
    orthonormal
    sets, Bessel’s inequality, complete orthonormal sets and 
    Perseval’s identity. Chapter nine contains an
    appendix giving a short 
    account
    of partially ordered sets and  Zorn's Lemma. Chapters eleven 
    and twelve discuss the Hahn-Banach theorem and
    its consequences including 
    the
    proof of the Riesz representation theorem for linear functionals 
    on  Hilbert spaces. In chapter eleven there
    is an appendix giving a 
    proof
    of the existence of ?nitely additive measures on all subsets of
    the 
    unit interval invariant
    for translation using the Hahn-Banach theorem-the 
    authors
    also prove there is no such construction for countably additive 
    measures. Chapter thirteen gives the Riez
    theorem for functionals on the 
    space
    of continuous functions on a closed interval. Chapters
    fourteen,?fteen 
    and sixtccn
    deal with the notion of weal; convergence, the space L{X, Y) of 
    bounded linear transformations of a normed
    linear space X into another such 
    space
    Y, the principle of uniform boundedness and some of its
    consequences. 
    Chapter sixteen
    introduces closed transformations, proves thc closed graph 
    theorem and the bounded inverse theorem.
    Chapters seventeen and eighteen 
    give
    an account of closure of transformations, completely continuous 
    transformations and conjugate (usually called
    adjoint) transformations. 
    Chapter
    eighteen defines the spectrum and resolvent set of :1. linear 
    transformation with illustrations. Chapter
    nineteen {the longest chapter in 
    the
    book — 43 pages) gives a short introduction to Banach algebras and 
    discusses the main results needed for use in the
    book. This chapter may be 
    the
    single most accessible introduction to the subject that exists
    anywhere. 
    Chapter twenty
    discusses properties of adjoint opcrators in a Hilbert space, 
    sesquilincar functionals and quadratic forms.
    Chapter twenty-one proves some 
    properties
    of the spectrum of normal and completely continuous operators on 
    a Hilbert space and has an appendix giving an
    account of tho Fredholm 
    alternative
    theorem. Chapters twenty-two to twenty-?ve discussing properties 
    of orthogonal projections, culminating  in
    the spectral theorem for 
    bounded
    self adjoint transformations on a Hilbert space. A second proof of 
    the same theorem is given in chapter twenty-six,
    and yet a third proof in 
    chapter
    twenty-seven. The same theorem for hounded normal opérotors forms 
    the subject matter of chapter twenty-eight. The
    last chapter gives thc 
    spcctral
    representation for unbounded self adjoint. transformations. There 
    are lots of good examples and strong exercises,
    including some optional ones 
    involving
    measure theory and complex analysis for graduate students using 
    the book. This makes the book quite flexible as
    a text. This book is a 
    classic
    and has earned its place as the quintessential introduction to the 
    subject-it will make the transition to more
    advanced functional analysis 
    books
    like Lax, Stein/Shakarchi, Reed/Simon or Yosida, 
    much easier.  It's a model of clarity and
    exposition-combined with 
    Taylor
    above, I can't think of a better introduction to graduate level 
    analysis. The highest possible recommendation!

  •  
  •  
  • Basic
    Algebra I Second Edition by Nathan Jacobson
    (PG-13/R)

  • Basic
    Algebra II: Second Edition by Nathan Jacobson
    : (PG-13/R)This
    is
    without question the single most
    important mathematics republication by

    Dover since the founding of the publishing house. I say we should nominate

    Dover Books for a Nobel Peace Prize for their
    recent reissue of this 
    classic.
    The late Nathan Jacobson, of course, was one of the giants of 
    non-commutative ring theory in the 20th century.
    He was also a remarkable 
    teacher
    with an awesome record of producing PHDs at Yale, including Charles 
    Curtis, Kevin Mc Crimmon, Louis H.Rowen, George
    Seligman, David Saltman and 
    Jerome
    Katz. His lectures at Yale on abstract algebra were world famous and 
    had 2 incarnations in book form: The first, the 3
    volume Lectures In Abstract
    Algebra,
    was for a generation the main competition for van
    der Waerden’s Modern Algebra
    as the text for graduate
    algebra courses. Basic Algebra is 
    the second major incarnation and this is a
    republication of the second 
    edition.
    The first edition came out in the 1970’s and was intended as an 
    upgraded course in algebra for the extremely
    strong mathematics students 
    entering
    Yale from high school during the Space Age. The first 
    volume-covering
    classical topics like groups, rings, modules, fields and 
    geometric constructions-was intended as a
    challenging undergraduate course 
    for
    such students. The second volume-covering an overview of categorical
    and 
    homological algebra as well
    as the state-of-the-art (circa 1985) of 
    non-commutative
    ring theory-was intended as a graduate course for first year 
    students. The complete collapse of the American
    educational system in the 
    1990’s
    has rendered both volumes useless as anything but graduate algebra 

    texts. Indeed, some of Jacobson’s former
    students tell me even most of the

    super-undergraduates at Yale struggled using the first volume as a first

    course in algebra after linear algebra. After
    several tries, Jacobson gave 
    up
    using it. He used both volumes until his retirement as first year 
    graduate course books with much
    better results. Still, given that the second volume was
    going for nearly 400 dollars at one
    point online in good condition
    , its 
    reissue
    by Dover in wonderfully cheap editions is a serious cause for 
    celebration. (I remember tearing my hair out
    trying to find a copy I could 
    afford
    online and when I finally did-I had to sell it to help pay for my 
    dad’s medical bills.)  Both books are
    beautifully and authoritatively 
    written
    with a lot of material that isn’t easily found in other sources, 
    such as sections on non-associative rings (my
    favorite)  ,Jordan and 
    Lie
    algebras, metric vector spaces and an integrated introduction to
    both 
    universal algebra and
    category theory. They are rather sparse in examples

    compared with other books, but the examples they do have
    are very well
    chosen and described
    thoroughly. There are also many fascinating, detailed

    historical notes introducing each chapter, particularly in the first
    volume.
    The main problem with both
    books is that Jacobson’s program here absolutely

    splits in half algebra into undergraduate and graduate level topics; i.e.
    without and with categorical and homological
    structures. This leads to 
    several
    topics being presented in a somewhat disjointed and inefficient 
    manner because Jacobson refuses to combine them in
    a modern 
    presentation-module
    theory in particular suffers from this organization. 
    Personally,
    I didn’t find it that big an
    issue with a little effort-but a lot of other students have
    complained 
    about it. Also, some
    of the exercises are quite difficult, rivaling Lang’s.
    Even
    so, the sheer richness of these books make them true classics. If 
    graduate students are willing to work a little to
    unify the various pieces 
    of the
    vast puzzle that Jacobson presents here with astonishing clarity, he 
    or she will be greatly rewarded by a master’s
    presentation and depth of 
    understanding.
    A wonderful graduate course could be taught using these books 
    and supplementing them with either Ash or the
    professor’s lecture notes. A 
    classic
    and a must have for any graduate student.

  •  
  •  

  • Basic Abstract Algebra: For Graduate Students and Advanced Undergraduates By Robert Ash (PG/PG-13)Ok, very quietly, this has become students’ secret weapon for their first year qualifying exams in algebra. It’s also the book that made the author nearly a household name in mathematics textbooks and it’s not hard to see why. The
    history of the book is remarkable-the book began life 
    as an extensive set of lecture notes Ash began compiling nearly 20 years ago for the graduate algebra sequence at the University of Illinois at Urbana-Champaign when he was frustrated looking
    for a textbook his 
    students
    could actually read and understand. He’d used Lang, Hungerford 
    and Jacobson for the course with mixed results.
    His students really hated 
    reading
    and working through any of them-especially Lang. Ash liked Lang’s 
    choice of material and thought with a little
    more concrete discussion and 
    explanation,
    it could be a terrific book. So he began writing a kind of 
    “supplemental guide” to Lang with many more
    examples and a lot of 
    rigorous,
    but concrete discussion. Over the next 2 decades, the notes
    evolved into the online version of the book and
    students all over the 
    internet
    found it to be very  illuminating-especially when used in 
    combination with Lang’s text. Gradually, there
    grew enormous demand for a 
    printed
    version of the notes and  Dover finally put out a nice, cheap 
    paperback and the rest is history.  It’s
    not hard to see why this 
    book
    in short order has become so beloved by struggling graduate
    students. 
    The book has huge
    coverage of the subject and a plethora of examples in 
    each section. (Indeed, there is a supplementary
    section to the first 4
    chapters
    that allows an instructor to use this part of the book as a stand 
    alone undergraduate text!) Groups, rings,
    fields, modules, algebraic 
    geometry
    and algebraic number theory(!) and much more. The book is 
    incredibly clear, comprehensive and illuminating
    with a style that is
    resolutely
    concrete but never lacking in either precision or rigor. The 
    real apex of the book is a nearly 60 page
    concluding chapter on basic 
    homological
    algebra that is one of the most lucid and detailed 
    presentations
    of the subject there is.  Frankly, if this was all 
    there was in the book, it’d still be worth
    having. But there is so much 
    more
    good stuff in here.  This includes brief sidebars into major 
    topics like p-adic analysis and group
    representation theory, complete with 
    references
    for further study- and Ash’s book contains complete, meticulous 
    solutions to all the exercises. There’s simply
    no more any student could 
    ask
    for in either a course text or for self study. This book is
    quickly 
    becoming a true
    classic in every sense of the word.  Ash has given a 
    huge boon to students of all levels and
    generations by making it available 
    so
    widely, easily and cheaply. My one quibble with the book is I wish
    he’d 
    included a more detailed
    section on group representation theory- that 
    subject
    still doesn’t have a good hard copy introduction in our price 
    range. Oh well-maybe in the second edition. We
    can hope.

  •  
  •  

  • Algebraic
    Topology by Allen Hatcher
      (PG-13)Rather shockingly and sadly, I actually
    had to bend the rules a bit to allow Hatcher into the list. But
    in the end, I decided the book was too important to leave off the list
    at the current price. The price of the book has steadily risen since
    it’s  publication 11 (!)
    years ago-I believe a new copy of the softcover will cost you 45
    dollars at this writing  on
    Amazon.
    Fortunately,
    Hatcher has kept the book freely available for download at  
    his  website and there are many, many used
    copies floating around for much 
    less.
    So either way, the 99 percent student can get a copy. The question 
    is-should he or she get one? Sigh. I’m kind of
    on the fence about that. 
    Like
    all students who have learned algebraic topology since the book
    was 
    published, I learned the
    subject from Hatcher-or at least, tried to. It 
    seems
    to have become one of those books that has developed almost a 
    religious reverence or revulsion in mathematics
    students and 
    mathematicians.  Dennis Sullivan
    absolutely loves the book and has said it was the book he wishes 
    he’d had when he was a student at Princeton
    suffering to learn the subject 
    from
    Norman Steenrod, who took much more of a formalist approach. Peter 
    May likes the book, but he thinks it’s too
    verbose and sloppy for a 
    graduate
    course and has used it in undergraduate courses at The University 
    of Chicago.  (
    Personally, I think that says more about the incredible caliber of
    mathematics students 
    at the U
    of C then of the difficulty level of the book! )  What 
    do I think of it? Well, to be honest, I have
    mixed feelings about it.
    Hatcher
    was trying to create a completely modern course on the subject 
    that first and foremost was about the
    geometrical roots of the subject. A 
    very
    laudable goal indeed that mathematicians have been trying to
    produce 
    since Edwin Spanier
    wrote his infamous book in the mid-1960’s and showed 
    to
    an entire generation of aspiring textbook authors how not
    to 
    write one. Did he achieve
    it? Well, yes and no and that’s the problem. 
    Yes,
    his emphasis on intuition and geometry is wonderful and is the
    book’s 
    real strength. The
    explicit constructions of the homotopy and homology 
    groups
    as well the accompanying dozens of examples and diagrams gives the 
    book an enormous richness and clarity very few
    books possess. Hatcher 
    concurrently
    expresses all this geometric content in completely modern 
    language-categories, functors and commutative
    diagrams are introduced
    early and
    used throughout in conjunction with the graphical content. So 
    what’s the problem? The problem is that Hatcher
    is so intent on focusing 
    on
    the geometric conception of modern algebraic topology, he tends to 
    neglect the basic tenants of precision in a
    mathematics textbook. He tends 
    to
    skip a lot of details in both proofs and examples and sometimes
    the 
    definitions can be hard to
    find in the prose. This makes the exercises, 
    which
    are written in the same style, almost inscrutable sometimes. Yes, 
    granted, this is a graduate textbook and you
    want to make the students 
    fill
    in details. But the problem is that there are so many new concepts 
    and methods that need to be digested in
    algebraic 
    topology that unless
    one is very clear in labeling things, it’ll be very 
    easy
    for a beginner to lose the forest for the trees. This is why
    although 
    I give the book a
    “thumbs up”, a more formal or better organized second 
    source-such as Joseph Rotman’s An
    Introduction To Algebraic Topology 
    or
    one of the excellent lecture note sets cited at the Lecture Note 
    Library-would assist the beginner in not getting
    lost in Hatcher’s style.  
    Those
    caveats aside, this is still probably the most accessible
    introduction to modern algebraic topology that
    currently exists. Here’s 
    hoping
    for a second edition-which Hatcher apparently has begun 
    planning-that gets the bugs out of this
    otherwise exemplary text.

  •  
  • Tensor
    Analysis on Manifolds by Richard L. Bishop and Samuel I. Goldberg
     (PG-13)This is
    one of 2 classics definitely worth considering as a general all purpose
    study textbook for graduate differential geometry issued by Dover (The
    other is Flanders). It’s been lurking on shelves for over 4
    decades, waiting for the curious student of differential geometry
    to crack it open and discover the many marvels waiting therein.
    Published in 1968 to rave reviews but little popular attention and
    then reissued by Dover in a corrected 1980 edition, this has been
    one of the line’s best sellers and for very good reason. (I’d
    actually be interested in asking Bishop or Goldberg about that book
    and what brought about its writing and what courses it developed
    out of.)  It’s basically
    a course for a first year graduate/ advanced strong undergraduate
    course in modern differential geometry. It covers all the standard
    stuff- differential manifolds, atlases and charts,
    coordinate systems, Lie groups and algebras and Riemannian and
    Semi-Riemannian manifolds. There are 2 unique aspects of the text.
    First of all, the presentation emphasizes tensor analysis from a
    completely modern perspective rather then differential forms. Forms
    are presented as tensor fields of a specific kind i.e. a skew
    symmetric smooth tensor field of degree p. Chapter 4 contains the
    most complete presentation of the linear algebra of tensors and the
    exterior product I’ve ever seen in a differential geometry textbook  This chapter alone
    makes the book worth having. This unifies the presentation and
    emphasizes the use of local coordinates, so critical in
    applications. This brings us to the second difference between it and
    other introductions to modern differential topology and geometry:
    the book closes with an entire chapter on the applications of
    tensor analysis to mechanics. The result is a rigorous mathematics
    textbook that it written for both mathematics and physics students.
    This was nearly unthinkable in 1968, the height of the
    Bourbaki era, when Western mathematicians would never dare let
    their diagram chasing be dirtied by any association with physics.
    Fortunately, the authors were wiser and more broad-minded and
    didn’t fall victim to the fad. If I had one complaint about the
    book, it’s the same one I have with most textbooks written in this
    period-that there aren’t nearly enough examples. The ones they have
    are good ones, but they are precious few. But other then that, this
    is one of the very best books for either mathematics or serious
    physics students to learn either tensor analysis or
    modern differential geometry and it’s one of the best Dover’s
    chosen to republish over the years. 

  •  

  • Differential
    Forms with Applications to the Physical Sciences by Harley Flanders
    :(PG)  This amazing book is
    considered a classic by physicists and it was the book most
    physicists, 
    engineers and
    applied mathematicians trained after the 1970’s cut their 
    teeth on when it came to learning differential
    forms and their
    applications.  I think most
    mathematicians, no matter how pure, should make it required reading
    by 
    all students who want to
    learn modern differential geometry, even if 
    it’s
    just as supplement to a standard text. The beautiful prose by 
    Flanders alone makes it worth reading-but
    there’s so much more here that 
    makes
    it a must-read. Granted, he’s not 100 % rigorous and careful in 
    the proving and construction of all results,
    but he’s not trying to be 
    and
    he says so in the  introduction.  What
    Flanders is trying to create here, like Bishop and Goldberg did
    for 
    tensor analysis above,
    is a book on differential forms for everyone-a 
    kind
    of compromise course to reconcile the long-lost brethren of pure 
    mathematics and its applications, whose
    absence from each other had 
    rendered
    both the much poorer in the 1960’s. His presentation is rigorous,
    but not so completely, anally so 
    that
    it alienates those trained in more empirical, intuitive thinking.
    Simultaneously, he provides much needed
    intuition and visual content to 
    the
    purely algebraic constructions of the Grassman algebra and forms.
    For example, I love how he demonstrates in the
    first chapter that the 
    exterior
    product is a straightforward generalization of the determinant 
    from undergraduate linear algebra to abstract
    linear spaces/ Later 
    chapters
    are loaded with both applications of all kinds-from surfaces in 
    Euclidean space to ordinary and partial
    differential equations to 
    mechanics
    to electromagnetism to relativity- and deep theorems relating 
    the multilinear algebra of differential forms
    to these applications and 
    their
    derivations. He has some of the clearest examples of commutative 
    diagrams I’ve ever seen! Flanders has hit on
    just the right combination 
    of
    abstract proof and intuitive examples. In short, the book is a

    treasure trove for both pure and applied
    mathematics students and it 
    would
    be required reading even if it didn’t have a low cost edition. 
    Since such a low cost edition does exist, it
    would be a crime not to 
    have
    one. I’m itching one day to try and teach a first year graduate 
    course in differential geometry from Flanders
    and Bishop/Goldberg!

  •  
  • Introduction
    to Logic by Patrick Suppes
    (PG)

  • Axiomatic
    Set Theory by Patrick Suppes
    (PG)The late Patrick Suppes
    was a well-known philosopher of both science and mathematics
    at Stanford University, as well as one of the lesser known founders
    of the computer science empire in Silicon Valley. He had one of the
    most eclectic and bizarre backgrounds of anyone in any of the 3
    named disciplines-I strongly recommend both his obituary
    and his personal autobiography at his web site for fascinating
    reading. What comes though in his life story is that Suppes learned
    early on the value of strong teaching and why this needs to be
    encouraged in our academic environments. This belief comes through
    very clearly in both of these textbooks, which were designed to be
    introductions to logic and formal set theory respectively, for all
    kinds of students with strong basic mathematical backgrounds.  And that’s
    really what’s so unique about Suppes’ books and why-even before they
    were available in Dover-they were so highly cited and used:
    Suppes was writing introductions to the foundations of mathematics
    for anyone who wanted or needed knowledge of these subjects.
    He’s very rigorous, but he’s careful to stay away from topics that
    are either purely philosophical or purely mathematical. I think a
    very good observation of how to describe Suppes’ logic text is made
    by John Myhill in his original review of the book in American
    Mathematical Society Reviews
    :

  • One
    can distinguish at least three attitudes towards the
    increasingly 
    important
    role of logic in the undergraduate mathematics curriculum; 
    the reactionary attitude which denies it
    any place; the moderate attitude 
    which
    regards it as a "luxury" subject, to be made available to those
    advan
    ced students who
    are especially interested; and the progressive attitude which 
    regards it as one of the earliest and
    most basic skills which a major should learn

  •  

  • Myhill
    places the book as a huge asset to the 
    progressive
    camp as it is broad, yet reasonably short and simple for a 
    beginner, be that student of either mathematical
    or nonmathematical 
    background.
    I completely agree-this is a book I wish I’d had way back when 
    I was originally a philosophy major taking logic
    for the first time. It’s 
    also
    a book that would be great reading in the summer months before a 
    serious first course in mathematical logic or
    set theory. Axiomatic set
    theory,
    in particular, really needs some familiarity with logic in order 
    to be fully comprehended. Trying to express
    axiomatic containment 
    sentences
    purely in ordinary English can be quite confusing. Which brings 
    us to the second classic in the pair from
    Suppes. This is a comprehensive 
    and
    crystal clear presentation of axiomatic Zermelo-Frankel set theory
    for 
    either strong
    undergraduate/ first year graduate students in mathematics 
    or advanced graduate students in philosophy.
    Despite the preface insisting 
    that
    a familiarity with logic isn’t assumed and that a very brief
    overview 
    is given in the
    beginning, it’s pretty clear that with a complete

    ignorance of the basics, this book is going to be very confusing
    sledding 
    for any student. But
    for students with a good grasp of the basics as set 
    forth
    in the first book, this book is going to be a joy to read, with 
    enormous depth, good problems and a focus on
    real-world uses of set theory 
    as
    the foundation of mathematics. A look at the contents shows the
    wonder 
    of this book and why it
    has been so widely cited: CHAPTER 1. Introduction: 
    Set
    Theory and the Foundations of Mathematics. Logic and Notation.
    Axiom 
    Schema of Abstraction
    and Russell's Paradox. More Paradoxes. CHAPTER 2. 
    Formulas
    and Definitions. Axioms of Extensionality and Separation. 
    Intersection, Union, and Difference of Sets.
    Pairing Axiom and Ordered 
    Pairs.
    Definition by Abstraction. Sum Axiom and Families of Sets. Power 
    Set Axiom. Cartesian Product of Sets. Axiom of
    Regularity. CHAPTER 3. 
    Relations
    and Functions. CHAPTER 4. Equipollence. Finite Sets. Cardinal 
    Numbers. Finite Cardinals. CHAPTER 5. Definition
    and General Properties of 
    Ordinals.
    Finite Ordinals and Recursive Definitions. Denumerable Sets. 
    CHAPTER 6. Fractions. Non-Negative Rational
    Numbers. Rational Numbers. 
    Cauchy
    Sequences of Rational Numbers. Real Numbers. Sets of the Power of 
    the Continuum. CHAPTER 7. Transfinite Induction
    and Definition by 
    Transfinite
    Recursion. Elements of Ordinal Arithmetic. Alephs.

    Well-Ordered Sets. CHAPTER 8. Some Applications of the Axiom of Choice.

    Equivalents of the Axiom of Choice. Axioms
    Which Imply the Axiom of
    Choice.
    I found his presentation much clearer on the technical points then

    Enderton’s, which famously-and
    confusingly-is done in ordinary language.

    Some particular points: His use of Tarski’s definition yields a very
    clear
    treatment of finite sets
    and allows the building of the finite ordinals in

    an exceptionally clear way, despite needing some extra machinery.
    Similarly, the building of the cardinal numbers
    through a separate 
    operation
    clears up many confusing points. In short, both books are an 
    absolute joy to read and no beginner should be
    without both when trying to 
    learn
    the essentials of mathematical logic.

  •  
  •  Advanced
    Calculus by Schlomo Sternberg and Lynn Loomis, 2nd edition

    (PG-13)  The book was finally(!) reissued in  fall 2014 in
    an affordable paperback edition authorized by Professor Sternberg,
    who 
    apparently finally got fed
    up with the ton of nagging he got over the 
    years
    to republish it. For many years, this was the Great Dark Secret 
    Textbook on advanced calculus. It was a book
    that was, in its own way, 
    scarier
    then baby Rudin or Herstein for  undergraduates. Notorious for it's

    level of difficulty, it was also the book
    first year graduate students in

    mathematics would make secret deals in the night for sums they had to
    sell
    their cars to get a hold of.
    It was the book you had to study to really

    master calculus of several variables. The 1990 edition was a rare gem
    that
    was selling for 300 dollars
    in fair condition online at one point. (I

    know-I had to sell my copy to help pay for groceries before my father

    passed. That one hurt far worse then
    selling Jacobson for the same

    reason.) When a scanned PDF version became available for download for
    free
    at Sternberg's website at
    Harvard, celebrations were held among

    undergraduates and graduate students.It was a huge gift to all
    mathematics
    students of all
    levels. It is a course on calculus on Banach spaces with

    examples and exercises that later stimulated research theses for the

    students who survived it. This book was
    written for the Math 55 course at

    Harvard in the late 1960's, an honors course in advanced calculus that

    justifiably has struck fear into the hearts
    of mathematics majors for

    generations as the most difficult undergraduate mathematics course in
    the
    United States. As far as I
    know, this is the only textbook ever written

    that really approximates the pace and coverage of the actual course.

    Reportedly, the course has been somewhat
    reduced in pace and level since

    this book was written. Apparently, the original version of the course
    was
    too much even for 17 year old
    full scholarship students that learned

    calculus in grade school that were arrogant enough to sign up. It was an

    academic meat grinder that left at most 3-4
    students standing at the
    course's
    end-many of which went on later to become faculty at the top tier

    universities in the world. Still, it's
    unimaginable that they actually

    taught-and to a lesser extent, still do-undergraduates this material at

    this level and superliminal pace. (I'm told
    by friends talented or foolish

    enough to attempt the course that it's really the pace of the course

    that's lethal more so then the level. If
    the course was spaced out more

    over 3 semesters instead of 2, it wouldn't be such an academic suicide

    mission.)  Then again,these were honor
    students at Harvard University
    in
    the late 1960's-argueably the best undergraduates the world has ever

    seen. In any event,for mere mortals,this is
    a wonderful first year
    graduate
    text and probably the most complete treatment of classical
    analysis on topological vector spaces that's
    ever been written. It even
    ends
    with an abstract treatment of classical mechanics. It's well worth

    the effort-but boy,you better make sure you
    got a firm grasp of both
    advanced
    calculus/ honors calculus/ undergraduate analysis of one variable

    and linear algebra first.  Kudos to
    Sternberg for making it available

    again to a broad audience, but I hope in addition he continues to make
    the
    free version available at his
    website. Still- very good news for all of us

    indeed. Very highly recommended-if you dare.

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