T.U.L.O.O.MATH-The Universal Lyceum Of Online Mathematics!!

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 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 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 19^th 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 R³  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 Rⁿ  

•     The book has a “spiral ascent” structure-it begins with the simplest kinds of functions of several variables, namely  the real valued maps of R² and 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 Rⁿ is 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 R³ , 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
  I 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
  advanced 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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