Any subject looked at from a historical perspective becomes interesting because the narrative becomes a story and the concepts become that much more meaningful. A subject like real analysis is a dry subject, whose importance though is seen in many branches of mathematics. When someone brings out a book on Real Analysis in a narrative format, I think it should not be missed. David Bressoud wrote the first edition of the book titled “A Radical Approach to Real Analysis” in 1994 and followed it up with a second edition in 2007. Obviously things have been structured/pruned/organized better in the second edition. But somehow I happen to read both the editions. So, I will attempt to summarize the content of both the editions as each has its own charm.

**Chapter 1 : Crisis in Mathematics**

An elementary view of a function is that it needs an input and it gives out an output. Considering this simple definition of a function, one usually visualizes functions as smooth curves / a set of piece wise collection of curves. Before the 19^{th} century, Functions were considered as polynomials, roots, powers, and logarithms; trigonometric functions and their inverses or whatever that could be built using addition, subtraction, multiplication, division or composition of these functions. Functions had derivatives; they could be expressed as Taylor series.

Four days before Christmas of 1807, Joseph Fourier submitted a manuscript which examined the flow of heat in a hypothetical rectangular plate. Everything was revolutionary about Fourier’s work. His solution showed that a constant function can be approximated as an infinite cosine series.

The above function challenged the world view of mathematicians. Term by term differentiation of this function gave rise to even stranger results. One could see that the series converged everywhere in the interval (-1,1) but term by term derivative failed to converge. Euler, Lagrange, Sylvestre Francois Lacroix , Gaspard Monge all were challenged to reconsider their view of function. However they resisted the change. This was to change in the 19^{th} century , thanks to Bernhard Reimann and Karl Weierstrass who sorted out the confusion about “What a function meant ?” and “What were the properties of a function?” This introduction of Fourier series at the beginning of the book shows the shaky foundations on which mathematicians/scientists during the early 1800’s were basing their work. Functions, the type of continuity they exhibited, differentiability of the functions, integrability of the functions were the concepts that had to be completely revamped, thanks to Fourier’s paper. Personally, I found this chapter to be the best introduction to the real analysis that I have come across till date. You look at Fourier Series and all the intuitive fundas about continuity, sequences, convergence, differentiability, integrability become shaky and you feel the need for rigorous understanding of the subject.

**Chapter 2 : Infinite Summations**

Since I happened to go over the first edition of the book, I have noticed that Chapter 2 has been revamped considerably in the second edition. A quick poll of the book amongst the real analysis enthusiasts reveals that some people love the first edition of the book while some people seem to like the second edition. I have found that reading first edition and then second edition was the best thing to happen to me. In the first edition there are a lot of topics covered but there seems to be areas where things are little unstructured. This is applicable more so to the second chapter of the book. But reading things at random sometimes helps you appreciate the value of structured content in the subsequent editions. Let me summarize chapters from both the version and you can see the difference between the 2 editions.

In the second edition of the book, the chapter starts off with Archimedes exhaustion method used to find the quadrature of the parabola.

The method used by Archimedes to find the area under the parabola is called “Method of exhaustion”, where a series of triangles are constructed and the area of the parabola is approximated by the summation of area of triangles. Here is what is revolutionary about his approach and that which shaped modern calculus.

When he came across this summation, he used a superb argument that whatever be the number k that is chosen the area can never be greater than 4/3 and whatever margin of error you are comfortable with, there is always a k for which the area will be in that margin of error from the true value of 4/3. By using this wonderful logic that infinite series should be viewed from the lens of partial sums, Archimedes method became widely popular and was applied to various concepts relating to functions.

In 1821, Cauchy used Archimedes argument to publish a book on infinite series where he showed the equality sign in infinite series has a different interpretation.

This is an important phenomenon. Ordinary equalities do not carry restrictions like this. The book then follows the content of the chapter 2 – Edition 1. So, the new edition basically places the Archimedes method as THE most important idea which lead to the further development of calculus by Cauchy and mathematical community.

Subsequently, the chapter explores infinite series and the way they are mystically different from finite series. The use of + , – , = in the context of infinite series has a completely different connotation. The development of the concepts around infinity came from search for better approximations of pi. Taking John Wallis approximation of pi/4, Newton generalized Wallis’s integral and came up with Binomial series, expansion of sine and cosine as an infinite series. So , these were starting signs of to use infinite series representations for known constants/functions. Euler’s constant was another breakthrough which used Nested Interval axiom and consequently infinite series to compute gamma, one of the universal constants known to us. The use of Taylor series exploded as they were smooth, continuous and infinitely differentiable. d’Alembert was the first to question the convergence of binomial series, though he did not pursue it further. Lagrange was the first mathematician to answer questions about convergence and divergence of the series using specific test criterion. Lagrange also formulated derivative concept using the terms in Taylor Series. This view was demolished by Cauchy with a counter example, which questioned the very basis of Taylor Series expansion where two functions have the same Taylor series expansion. We are no longer talking about the accuracy of the Taylor series expansion of the function but we are interested in whether the Taylor series converges to the actual function and whether it makes sense to talk about Taylor series expansion for any function.

**Chapter 3 : Differentiability and Continuity**

In the first edition of the book, the chapter starts off with the well known Newton Raphson method and shows that the method does not work for all initial values. There are functions and specific intervals for those functions where Newton Raphson works chaotically. Through a wonderful example, one sees the reason for Newton Raphson working for only a specific range of starting values. Now one might wonder the reason for using this method as an example at the beginning of a chapter on differentiation. However in the second edition, this example is pushed to appendix and the reason is stated clearly at the beginning of the chapter , Cauchy’s concept of derivative is tightly linked to Lagrange error test , which depends on the differentiability of the function. Newton Raphson does not behave as needed for a certain start values because there are intervals where the function is not differentiable.

Epsilon delta definition is given for specific functions and application of the definition is shown using several examples. Basically it’s a game of “you give me epsilon, I give you delta” that characterizes the method for verifying the differentiability of a function at a point. This epsilon-delta method clearly shows the reason for differentiation failing for some of the infinite series. One cannot differentiate Fourier series which has infinite terms in a simple fashion by differentiating term by term. There are few examples cited which show the importance of using epsilon delta approach, where the standard rules of differentiation would not help you find the derivative.

So, to proceed any further, Cauchy realized that it was necessary to rigorously prove Legendre’s Remainder theorem for n= 1, which is popularly known as “Mean Value Theorem”. Cauchy’s attempts to proving mean value theorem is provided which gives the reader a chance a look at the humungous efforts that were undertaken to form a rigorous definition of continuity .

First proof had few major flaws where Cauchy assumes boundedness for the derivative and also assumes a fixed delta works for all the points in the interval. Cauchy’s second proof had a circular argument and hence did not give a thorough understanding of continuity.

It was Bernhard Bolzano (1917) who was credited for the appearance of the modern definition of continuity. He discarded the use of Intermediate Value Property as a prerequisite for the proof. The book then introduces the epsilon delta definition of continuity which basically implies intermediate value property, boundedness and other relevant properties. This approach is different from Cauchy’s approach where intermediate value property was used to prove mean value theorem. There is also a mention of Dirichlet and Thomae functions where the former is nowhere continuous in R while the latter is continuous at every irrational point of R and discontinuous at all rational points of R. The chapter then deals with the offshoot of continuity property. Continuity is a powerful concept and many properties flow from this continuity aspect. If a function is continuous on [a,b] then it is bounded. It also achieves its greatest lower bound and least upper bound. Rolle’s Theorem, Mean Value theorem are subsequently stated for continuous functions.

**Chapter 4 : The Convergence of Infinite Series**

What does an infinite series mean and how can we manipulate ? Well, if one were to work on such mathematical objects, one has to change one’s view to make progress. It is not the terms that we should look at but the partial sums. The sequence of partial sums gives us a way to use epsilon delta approach and thus one can talk about the convergence or divergence of the original series. The fact that the original series can be retrieved from the partial sum series is the reason for shifting the view to partial sums and working on them. In simple words, the partial sums play a defining role in the formulation of value that is eventually assigned to an infinite series. The way we check the convergence of an infinite series is to create error partial sums and then play the old game of epsilon-delta. In this case, it is an epsilon-N game: given an error bound epsilon, and one has to find a response N so that whenever one takes at least N terms of the infinite series, the discrepancy between the partial sum and the value assigned to the infinite series is less than epsilon. If there is a response for every positive error bound, then the infinite series has this value.

One aspect that is radically different for an convergent infinite series is that , the usual equality you see in a sequence as simple as 1+1/2+1/4+1/8 +… = 2 does not mean the LHS equals RHS. How many ever terms you take on the LHS, it will never become 2. It will become closer and closer to 2. So, how does one interpret = in the above equation. The = sign means that as n approaches infinite, the partial sum , Sn converges to 2. Basic tests of convergence are discussed. Merely summing the terms for a specific N might give a false sense of convergence or divergence. A few examples mentioned in this chapter drive home this point.

Cauchy, stands out as a towering individual who first addresses the question of testing convergence of a series whose convergent value is unknown(which is most often the case). His theorem talks about convergence by using the partial sums ONLY and plays the epsilon-delta game to show convergence or divergence of a series. Like how a Monotone Convergence theorem helps in deciding convergence of a function, Cauchy criterion is useful for Infinite sequences without the mention of the word “LIMIT”. There are other tests mentioned such as absolute convergence test, Comparison test, Ratio test, Root test, Cauchy Condensation test, Integral test which are all ways to check the convergence or divergence of a series. Hypergeometric series are introduced and the all powerful Gauss test is stated and proved. Convergence of a power series is also discussed and in this context Radius of Convergence is clearly explained. The chapter ends with proving the convergence of Fourier series , thanks to humungous efforts of Abel and Dirichlet.

**Chapter 5 : Understanding Infinite Series**

Infinite series are strange animals as the general rules like commutative property does not hold good for the series. Can one interchange summation and limit sign of a sequence ? Can one interchange the differential operator and summation operator for an infinite sequence? Can one interchange the integral operator and summation operation for an infinite sequence ? These questions are very important and they need to be tacked with precision. This chapter gives the ideas of Bernard Riemann who first introduced the concept of absolute convergence and conditional convergence. Cauchy tried to answer such questions but did not take in to consideration uniform convergence. This uniform convergence is then used to ascertain conditions under which the differentiation and summation operators can be interchanged, integral and summation operator can be interchanged etc. The chapter ends with describing tests to verify uniform convergence, Weierstrass M-test being the most prominent one.

**Chapter 6 : Return to Fourier Series**

This chapter deals with Fourier Series which has been mentioned at the beginning of the book. Cauchy’s and Riemann’s efforts finally lead to the proof that Fourier Series is continuous. This chapter is very dense and it might require multiple readings. I did not understand a lot of aspects of this chapter as it starts talking about continuous nowhere differentiable function and related aspects.

After reading this book, I was so fascinated by the way the book was written that I have placed an order for the sequel for this book which is on Lebesgue Integration. I just hope that the book on Lebesgue is as good as this book.

The historical context of Real Analysis is brilliantly written by the author. There are a lot of examples that are worth going through. Personally, I will forever remember the chapter on Fourier and its importance in shaping the course of calculus. The approach taken by the author is definitely radical and thus is a refreshing change from the usual dry real-analysis books that one comes across.

__For a perseverant reader, this book is a gem. __

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