For any budding mathematics major, there are several litmus test courses which provide critical skills needed for later research. Abstract algebra, linear algebra, and topology are all such “core” courses for anyone pursuing a degree in mathematics. Of all these courses, it is probably elementary real analysis which is the most difficult hurdle to overcome.

In my review of Walter Rudin’s Principles Of Mathematical Analysis *,* I lamented the deplorable state of calculus courses at most colleges, which I see as the primary reason the theoretical approach to calculus and analysis poses such a problem to mathematics undergraduates. I come from a rather strange background when it comes to real analysis: I am mostly self-taught in the subject. I learned real analysis primarily from working through the deep and beautiful two-semester course notes of Gerald Itzkowitz at Queens College while auditing his course, supplemented by Kenneth Ross’ Elementary Analysis: The Theory Of Calculus and Tom Apostol’s ven erable *Mathematical Analysis* . (A detailed study of Rudin’s text came later. In retrospect, I was fortunate in this regard.) In the process, I sat and studied with many of my fellow students — for most of us, it was a struggle. Working with inequalities and the properties of the real numbers and making approximations rigorously for the first time using those properties is like learning a strange new language to most students. At times, analysis seems like an arcane, impenetrable bag of magic tricks performed to the initial mystification of the captive audience of students, who are then expected to learn tricks of their own. The familiar and friendly calculation devices from calculus suddenly become trivial byproducts of maddeningly abstract, complicated and alien constructions such as partitions and Cauchy sequences. This is a frustrating experience for all but the very best students. For such students, any help they can get, as cheap as they can get it, is a Godsend.

Which brings us to Dover’s recent reissue of Kenneth Hoffman’s *Analysis In Euclidean Space —*which may be the most significant addition to Dover’s distinguished line of blue-collar paperback texts yet.

If you’ve never heard of the text, that’s ok. It’s shocking that outside of the Massachusetts Institute Of Technology, very few people have. More surprising is that the book didn’t really have a lot of competition when it was first published in 1975. The standard real analysis text was of course Rudin, then in its 2nd edition for a dozen years (the 3rd edition would be published the following year). The 2nd edition of Apostol had been published the year before and was rapidly gaining the enormous popularity it still enjoys today. There were a *few* other texts — Fulks’ *Advanced Calculus*, Mardsen’s *Elementary Classical Analysis, e* tc. But most of these were considered also-rans: Rudin and Apostol were the only texts of choice for serious math majors.

Hoffman’s book really should have made more of an impact in that climate, especially given who its author is — Hoffman is 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 have caused more of a buzz. I learned of the text only because it’s mentioned as a reference for the undergraduate real analysis prerequisites in the preface of the first edition of Richard Dudley’s *Real Analysis And Probability* . I certainly knew who Hoffman was well before that: My mentor Nick Metas knew him well as a graduate student at MIT in the early 1960s as he was part of the team of such students who worked on the lecture notes from which Hoffman and Ray Kunze developed the early draft versions of their classic text *Linear Algebra* . The book under consideration developed out of Hoffman’s lectures on undergraduate real analysis at MIT beginning in the late 1960s. Hoffman had a reputation among the students at MIT as passionate teacher and communicator of mathematics. (Apparently he still does, as he’s been quite active in the mathematical community after his retirement.) Judging from this wonderful and unusual text, that reputation was well deserved.

Hoffman’s interest in teaching is clear from jump: we find not only the usual preface for instructors and reviewers, but aso a special preface addressed to the student with many helpful suggestions. Chapter 1 begins with the expected development of the properties of the real numbers. There are two ways to present this in an introductory analysis text — either you can present a brief construction of the reals or you can simply take the properties as axioms and build the structure of analysis from there. Hoffman comes down firmly in the latter camp, but he presents many examples and he explains in considerable detail *why* these axioms are needed to preserve the usual properties of the real numbers. This is indeed typical of the style of the book — the focus is much more on *why* than *what*.

There a number of very atypical discussions in this preliminary chapter, the most important of which is a detailed review of the vector space properties of **R**^{n} . This is highlights the single major difference in the text as opposed to the usual presentations: 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 **R**^{n}.

This is a little jarring, but Hoffman makes a good case for it in the book. 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.

Chapter 2 begins analysis proper, with convergent sequences, infinite series and their related limits and properties, as well as compactness and connectness for subsets of **R**^{n}. This shows another advantage of Hoffman’s approach — compactness, connectedness and sequential convergence are more clearly related to the geometry of **R**^{n} than to that of the real line, and this can be used to give geometric examples. Relying heavily on vector space properties of **R**^{n} makes the material unusually visual without sacrificing any rigor. Chapter 3 discusses continuous functions, limits of functions and uniform continuity in **R**^{n} , again relying heavily on the linear properties of **R**^{n} . For example, Hoffman discusses rigid motions as an example of uniformly continuous maps — he also gives a brief but deep discussion of the complex exponential on the unit circle using only convergence and linear algebra.

Chapter 4 gives an overview of calculus armed with the concepts of analysis needed to fully rigorize its concepts: differentiation on **R**^{n} described on both vector functions composed of real valued coordinate maps and directly as linear maps in **R**^{n}, the extreme value theorem and mean value theorem, the chain rule and convex functions. The bulk of Chapter 4 consists of a careful development of both the Riemann and the Riemann-Stieljes integrals in **R**^{n} in terms of suitable partitions and their meshes as well as upper and lower Riemann (Stieljes) sums and functions of bounded variation. The development here is particularly careful and clear with detailed proofs; Hoffman shows there are several possible ways to develop these concepts and some are better then others depending on purpose.

Chapter 5 describes sequences of functions, pointwise and uniform convergence and equicontinuity of these sequences in **R**^{n} . This presentation is for the most part standard although it contains considerably more detail then either Rudin or Apostol; most significantly, multiple sequences and series, complex analytic series, the Cauchy analyticity criterion, and the Moore-Osgood double convergence theorem are discussed.

Chapter 6 is really where the text breaks from the mold as it gives a very detailed discussion of the structure of general normed linear spaces; Hoffman shows most, but by no means all, of the geometric and analytic properties of Euclidean space do hold in such spaces. This chapter is clearly designed not only to generalize the results of the previous chapters, but also to prepare mathematics majors for follow-up courses in functional analysis. Hoffman discusses semi-norms and semi-normed spaces, subspaces, the sup norm, the general Cauchy-Schwarz inequality, Lipshitz norms and continuity, completion and complete normed spaces, Hilbert and Banach spaces and the basic properties of Fourier series, along with many, many examples from geometry, linear algebra and calculus. The culmination of this unique chapter is a discussion of quotient spaces and the completion of a normed space — this leads to a proof of the fact that every normed linear space is a dense subspace of a Banach space!

Chapter 7 continues this unique path with a detailed presentation of the Lebesgue integral à la Daniell, focusing on functions rather than measures. Using the machinery of general normed spaces developed in the last chapter, Hoffman defines compact support, weak continuity and fast Cauchy convergence, and then proceeds to develop the Lesbegue integral and its main theorems in detail, including a full discussion of general multiple integrals and the Fubini theorem, ending with orthogonal expansions. However, this list is deceiving, as Hoffman also gives a great deal of historical and intuitive discussion on general measures and their properties without beginning with them as the elements of developing the general theory — this will give students a lot of insight into why this concept was so important in the development of analysis. Also, a great deal of the traditional “measures first” approach is given as a series of challenging exercises for the student after seeing the “functions first” approach — Hoffman is careful to emphasize the countable additivity of measures and sets of measure 0 in order to prepare students for this task.

The final chapter gives a rigorous presentation of general differentiable mappings in Euclidean spaces as affine transformations on such spaces. Change of coordinates, the Inverse and Implicit function theorems are also discussed emphasizing again the properties of linear spaces and invertible linear transformations. The book concludes with an appendix on set theory, functions and cardinality — all with the author’s usual clarity.

Hoffman isn’t scared to challenge the reader in the problem sets. Analysts often complain that the exercises in Apostol's book are soft compared with the hard edged exercises of Rudin. Hoffman’s problem sets are far from soft without sacrificing an ounce of detail or clarity of presentation. So it *can* be done.

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. 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. There are many, many pictures — the geometric perspective makes this very natural and not forced as in many analysis texts. Hoffman’s approach allows one to build upon the well-prepared students’ already fluent understanding of the geometry of such spaces — the emphasis on the geometry of such spaces is unique also for a real variables text.

There are “true and false” questions in the exercises — questions that really require a great deal of thought on behalf of the student. A few examples:

**(3. 6 Exercise 6):** Let F: D → **R**^{n} where D is the unit circle in the plane and S is a subset of **R**^{n}. True or false: If F is uniformly continuous and S ∩ D is bounded, then F(S) is bounded.

**(6. 5 Exercise 3)** If you did not know Ascoli’s theorem would you regard it as obvious that any sequence of smooth functions on [0,1] with derivatives bounded by 6 has a subsequence which converges uniformly on [0,1]?

The exercises and examples int his 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.

Hoffman’s *Analysis On Euclidean Space* is a forgotten classic and its reissue in this beautiful cheap edition is a cause for celebration for all lovers of mathematics from Harvard to Hunter. I would love to assign this as the text the first time I teach real analysis. Instructors who do this will be doing their students a great service — not only educationally, but financially. Its’ a book whose depth is such that *everyone* can learn something from it — doesn’t matter if you’re an honors freshman struggling with epsilon-delta arguments for the first time or you’re doing your PhD thesis in operator theory at Yale. Buy your copy now — you’ll be glad you did. I know I am. And we can all thank Dover for making it available again.

Now if only we could get them to reissue Pierre Samuel’s *Projective Geometry*…

After many years in purgatory, Andrew Locascio is now a master’s student in mathematics at Queens College of The City University Of New York. He is a firm believer in the organic perspective of mathematics, which states theory and applications are but 2 sides of the same coin, which implies the unity of all natural sciences under the umbrella of mathematics despite what the average medical student thinks. He is as old as his tongue and a little older then his teeth and beyond that his age will remain a mystery to all save his physicians. He is hoping for a small window of good health in between getting his PHD and dying in order to make a small contribution to human civilization.