### October 2010 Inequalities are a boon to various derivative valuation problems that arise in real life. One can use inequalities to get around problems for which closed form solutions are difficult. The other day I was struggling with an inequality . So, took my friend’s help who worked on it in a jiffy and produced a bound. He also happened to suggest this book to me so that , I can quickly refresh basic inequalities and probably stop bothering him with trivial questions –:) . This book is a remarkable little gem that quickly got me up to speed .

First three chapters of the book can be speed read as they introduce the inequalities from an axiomatic point of view. Chapter 4 is probably the best in this book as it states and proves arithmetic-mean-geometric-mean inequality, Cauchy inequality, Holder inequality, Triangle inequality, Minkowski’s inequality using algebra and geometry proofs.

Chapter 5 talks about various applications of the inequalities. Most of the applications use the Arithmetic mean Geometric mean inequality except one example where equation of a tangent is derived using Cauchy’s inequality( I liked this example as it was a very simple and cool application of Cauchy’s inequality). Chapter 6 is extremely relevant from a metric space perspective, where various inequalities are applied to prove that a metric has the relevant properties of distance metric. For a person looking to brush up his/her knowledge on inequalities, this book is apt. It can be easily read within a couple of hours.

Instead of approaching “Metric spaces” from a real analysis perspective, this book tries to use an application to motivate the reader. The application is based on “contraction mapping theorem” which is used in solving a single equation / simultaneous equations. In its most simple form, let’s say there is an equation of the type x = f(x) which needs to be solved. One of the ways to solve this type of equation is as follows:

Start with an initial value, x1, and then calculate x2= f(x1), x3= f(x2) , x4 = f(x3) , …. etc. The sequence x1, x2, x3,…. converges to the root of the equation.

Now the obvious questions are : Why does this procedure work ? When does the procedure break down ? The root of such an equation is called fixed point as function does not change the value. The principle behind this simple question can be found in the concepts underlying “Metric Spaces”.

Typically one comes across metric spaces in a real analysis text. This book is welcome breather to people who want to know about metric spaces but are rusty on “Real Analysis” fundas.  Let me quickly summarize the content of the book.

Chapter 1 : Sequences by iteration

The first chapter provides motivation to get a feel of the sequences and their convergence. With the need to solve a simple equation x= f(x), where f(x) is continuous, a few examples are shown where iterative procedure gives the root of the equation. The questions that are asked for each of the examples are :

• Will the sequence of x’s that result from iteration ALWAYS converge ? If not, what are the conditions under which the sequence does diverges?
• If it does converge , Will the limit definitely be a root of x = f(x)

Iterations need not deal with only numbers. If you take a function x(t) , and apply a transformation (let’s say an integral transformation) , it is likely that the transformation gives back x(t) . Hence there needs to be a framework to solve such problems where a transformation does not the change the function. Basically the first chapter is used to show the reader various examples that use a sequence of numbers / sequence of functions/ to solve fixed point/function problems. One lingering question at the end of this chapter is, “What does it mean to say that a sequence of values / functions converge?”

Chapter 2 : Metric Spaces

The second chapter introduces the concept of metric spaces. The basic definition of a metric d is given below: Metric space is then symbolized as (X,d). To give a general flavour of metric spaces, the following metric spaces are discussed

• X = R , d( x, y ) = |x-y|
• X = C,  d( x, y ) = |x-y|
• X = R^2 , d( x, y ) = Euclidean Distance
• X = R^n ,  d( x1, x2,…, xn ) = Euclidean Distance
• Discrete space
• X = R^2 and d( x, y ) is lift metric / raspberry pickers’ metric
• X is the set of continuous functions from [ a, b ] to R , d( x, y ) is based on max metric
• X is the set of bounded functions from [ a, b ] to R , d( x, y ) is based on sup metric
• X is the set of continous functions from [ a, b ] to R , d( x, y ) is based on adding up all the distances apart of their graphs.

The examples also show that it is important to think about mathematical objects in terms of their roles than their existential meaning, a point extremely well pointed out by Timothy Gowers in his short intro book on math. Once a metric space is defined with a certain properties, it allows one to talk about metric spaces in a variety of contexts.

In the chapter, subsequently there is a discussion on sequences and Cauchy convergence is discussed. One crucial but often neglected point mentioned at this point of the book is “ Whenever we talk about a sequence , we need to mention the metric space that we are taking about” . Reason being a sequence can converge for a specific metric in R while diverge for a different metric in R.

Chapter 3 : The three C’s

The third chapter is titled “The three C’s”, referring to Closed, Complete , Compact properties of a set.This chapter is a little dense but one needs to know these 3 C’s to appreciate Metric spaces as some properties of the sets are invariant under change of metric space. Compactness does not depend on metric spaces. The chapter introduces each of C’s from a sequence angle. The basic idea behind introducing these concepts is to answer the question, “If there are a series of numbers in a sequence, how can one talk about convergence ? ”. A look at Cauchy convergence does not guarantee convergence / uniform convergence. Hence there needs to be additional constraints to take a Cauchy sequence that results out of iterative procedure to call it the root of the equation. What are the three C’s in simple words?

A complete set if one in which if a sequence has the property that the distance between terms tend to zero, then the sequence converges to a point in the set. A complete set is necessarily closed but a closed set need not be complete. Why bother about complete sets? Because these are precisely the sets for which the Cauchy convergence is enough to ensure convergence. If we restrict the attention to complete metric spaces (X,d) then subsets of X are complete if and only if they are closed. The takeaway from this chapter is the following dependency

Compact => Complete => Closed

One of the other takeaways from this chapter is that, if a sequence is bounded and closed, it is compact.

Chapter 4 : The contraction mapping principle

This chapter talks about the underlying principles of the contraction mapping theorem. The first chapter is all good as it appeals to the intuition. However one needs to get the math behind it and this chapter explains using the concept of contraction. Using the above concept, if f be a contraction of complete metric space, then f has a unique fixed point. Another extension to the above definition is when the space is compact metric space. In such a metric space, a weaker condition would suffice, k can be equal to 1.

So, this is where all the concepts come together in the book. Metric spaces, complete sets, compact sets are beautifully tied to the framework for solving x=f(x) equation. The best thing about defining rules about mathematical objects is that it allows you to move in to various dimensions / spaces. The chapter then goes to show the concepts behind the contractual mapping theorem as applied to PDEs and integration involved equations.

Chapter 5 : What makes analysis work ?

The last chapter goes back to real analysis concepts introduced in chapter 2 and tries to reiterate the concepts of open balls, closed balls, uniform continuity, closed sets, compact sets, connected sets. It then applies these concepts to classic theorems in real analysis. So, the last chapter , I think serves as a motivator to the reader to go back to heavy real analysis machinery books like Rudin etc.

If you want to understand “Metric Spaces”  and are rusty with real analysis concepts, this book is perfect. The treatment is different in the sense that it avoids “epsilon/delta”  argument  and uses sequences to explain the principles behind “Metric Spaces”

Off late, I have developed a lot of interest in Lebesgue Measure, thanks to the math- fin exposure over the last few years. Being a practitioner instead of a theorist/academician has its own advantage. You don’t get wedded to one concept or one theory. You take a random sample of all the techniques which have been applied to solve a problem and based on the context, you can choose one  from the random sample OR create a customized method from that random sample of methods. Sometimes it also helps to see how various problems have been approached by the pioneers in any field. The Calculus Gallery is one such book which gives the approach followed by pioneers in Calculus to solve various kinds of problems.

My main interest though was the  chapter titled Lebesgue describing the  work done by Henri Lebesgue. However to appreciate the last chapter, it is better to go through the initial chapters of the book that document the trials/ failures/ breakthroughs of  various mathematicians /scientists over the past few centuries.

Calculus has had a very interesting history where various mathematicians, scientists, scholars have contributed to its development.  This book summarizes the main personalities who played a role in these developments. Let me attempt to summarize the towering personalities mentioned in the book. Issac Newton ( 1642 – 1727)
Mathematicians credit Newton as the creator of calculus. A physicist by background, he invents calculus to help his work. The book talks about Newton’s early achievements like generalized binomial expansion for turning certain expressions to infinite series, methods to invert such infinite series and quadrature rules. In his manuscript , “De Analysi”, Newton spelled out the rules for determining quadrature. In modern calculus jargon, his rules basically were relating to switching summation and integrand & using infinite series in the evaluation of integrals. The highlight of this chapter is the sine series expansion. Most of us , if asked to expand sin(x) would just use taylor series and produce the infinite series expansion. But that’s now how Newton visualized. Instead he derived a series for arcsine and then inverted that formula to develop sin(x) expansion.  Mathematics, as evident from this simple example has never evolved as portrayed by textbooks. It developed by fits and starts and odd surprises. Gottfried Wilhelm Leibniz ( 1646 – 1716)

Leibniz is also credited for inventing Calculus. There is a bitter debate about whether he stole it from Newton or nor. Well, that’s useless for us as what matters are his ideas which were splendid to say the least. Leibniz had an immense talent for coming up notations. His notation for integral and differentiation was one of the reasons for the spread of calculus in Europe as compared to UK (Newton’s notation of fluxions was not viral in nature). Finding area under the curve was the hottest topic during these times and author explains the Leibniz approach, a.k.a, the transmutation theorem. Also mentioned is the way in which Leibniz derived the infinite series expansion of pi. Besides these mentions, Leibniz codified and published his ideas on differential and integral calculus, thus popularizing it amongst the European Mathematicians and Scientists. Jakob Bernoulli ( 1654 – 1705) & Johann Bernoulli ( 1667 – 1748 )

If Newton and Leibniz were the architects of Calculus , the Bernoulli brothers were builders. They worked incessantly on infinite series, brought in terms such as “integral” in to the calculus vocabulary.Jakob dealt extensively with arithmetic, geometric, harmonic series and figurative series. Besides making immense contribution to the understanding of calculus, Johann Bernoulli also mentored Euler, regarded as one of the history’s greatest mathematicians.

Leonhard Euler ( 1707 – 1783 )

There is virtually no field of mathematics where he has not made contributions. Euler is often called the publishing genius  as he is credited to have written 18 thick volumes and nearly 9000 pages on just “analysis” alone. The book covers 5 aspects out of the humungous contribution by Euler. Euler was the first person to bring forward the ratio view of calculus. He was of the view that infinitesimal quantities by themselves did not mean anything and it is always the study of ratio dy/dz that is important. He also wrote 3 volumes of books where he formulated and proved all kinds of integrals. Instead of attacking one infinite series at a time, he came up with a solution which gave the result to a set of infinite series all at once. Perhaps one of the significant contributions of Euler that is evident in stats and probability is the gamma function. Some of the popular density functions have gamma function embedded in it and every stats researcher  stumbles on to it at some point or the other.

In spite of the contribution of the above personalities, calculus had shaky foundations. Most of the techniques used infinitesimals erratically. Integration of infinite series was done as though it was finite series. No justifications were provided.  In the meantime, another mathematician Lagrange offered a different view : Infinite series as a source of differential calculus. He formulated everything in terms of infinite series. Alas! a disaster struck him when Cauchy presented a counter example where two functions have the same infinite series. So, even after 200 years of developments in calculus, there was a need to work on revamping the entire field from its foundations to give it a definitive rigour.

Augustin-Louis Cauchy ( 1789 – 1857 )

Cauchy’s stamp on calculus is of gigantic proportions. He grounded intuitive notions of continuity and integrability by laying solid foundation. He first defined limit and then attempted to formulate continuity based on the concept of limits. One of the Cauchy’s achievement was to prove everything using rigorous analytic methods. Intermediate Value theorem which sounds intuitive to any reader , was first dealt with a proof for it by Cauchy. He assumes nested interval property and shows the proof. This is a classic case where pursuit of rigor brought in wonderful new concepts like Axiom of Completeness later worked out by other mathematicians based on Cauchy’s work. Cauchy also proved mean value theorem and elevated it to the a place where it became central to the development of calculus. Before Cauchy, there was a view was integral was anti-differentiation. However Cauchy changed that view and believed that integration should exist by itself . His revolutionary idea was that integral should be linked to the limit rather than the continuity of a function. He then worked to show the connection between differentiation and integration, which is popularly known as the Fundamental theorem of Calculus. However these herculean efforts/ proofs had a few wrinkles like his assumption of uniform continuity of the functions. This and other wrinkles were addressed by Riemann.

Georg Friedrich Bernhard Riemann  ( 1826 – 1866 )

Riemann’s provocative idea was to divorce integrability from continuity. He pursued a solution to the question,” What does definite integral mean ?“. His approach is probably taught as the first lesson in integration across most schools. The concept of Riemann Sums was introduced and integration was tightly linked to the convergence of the infinite series of Riemann sums.   More importantly, he showed that Dirichlet function was not integrable, a breakthrough in calculus. He also gave an example of a discontinuous function that was perfectly integrable thus raising a critical question,” How discontinuous can an integrable function be ? “

Joseph Liouville ( 1809 – 1882 )

In the context of calculus, Liouville is famous for proving that certain integrals have no closed form solution and proving that transcendental numbers exist. The proof is covered in this book mainly to show the fantastic application of inequalities which was later adopted by Weierstrass to bring enormous precision in the definitions of limit/continuity/differentiation etc

Karl Weierstrass ( 1815 – 1897 )

The modern era would not arrive until the last vestige of imprecision disappeared and analytic arguments became for all practical purposes, incontrovertible. The mathematician most responsible for this transformation is Karl Weierstrass. He defied limit using epsilon delta notation with inequalities thus removing all traces of geometry in a definition. He also worked on the wrinkle in Cauchy’s proof – i.e assumption of uniform convergence. His major contribution came in the form 3 theorems. First relating to the convergence of a sequence of continuous functions, Second relating to the switch of integrand and the summation sign and third, the most helpful Weierstrass approximation theorem. He also came up with a mind blowing counter example of a continuous no where differentiable function which lead the subsequent mathematicians to make real analysis more rigorous.

So, are all the flaws worked upon/ theory complete / acme of real analysis has been reached ? Well of course not… A series of counter examples showed that the problems in the field were far from addressed. Thomae function showed that inspire of infitude of discontinuities, the function was integrable over [0,1] . There are three famous counter examples quoted in this book.

• First example shows that even though the function is discontinuous at the origin , it satisfies intermediate value property over every interval.
• Second example shows that “every where continuous function” is not the same as “ able to draw with out lifting the pencil”.
• Third example shows that a differentiable function can be discontinuous.

The functions thus raised important questions which lead to a completely different and radical development in the field of calculus.

• Can we construct a function continuous at each rational and discontinuous at each irrational ?
• How discontinuous can a Riemann integrable function be ?
• How discontinuous can a derivative be ?
• How, if at all, can we correct the deficiencies in the Riemann integral?

Calculus was in need of fresh eyes! Georg Cantor ( 1845 – 1918 )

Bertrand Russell described Cantor as “one of the greatest intellects of nineteenth century. Developments in the nineteenth century placed calculus squarely upon the foundations of limits. It had become clear that limits, in turn, rested upon properties of the real number system, foremost amongst which is what we now call completeness.  Cantor in his 1874 paper demonstrated that a sequence cannot exhaust an open interval of real numbers.  He then went on to define cardinality of a set and showed that there are different types of infinities. Cantor proved that a denumerable set, although infinite , was insignificantly infinite when compared to a non-denumerable counterpart.  This dichotomoes between large sets and small sets were also shown in other analytic settings.  Cantor is also famous for his continuum hypothesis which says that there is no set A which follows the inequality : Cardinality of N < Cardinality of A < Cardinality of R. Cantor’s distinction of large sets and small sets set forth interesting developments in the field.

Vito Volterra  ( 1860 – 1940 )

Vitl Volterra became famous for his work on pathological functions. He produced a counter example of a function which was differentiable everywhere, was bounded, yet was not riemann integrable. His other notable work was on showing that there is limit to this pathological functions, for example, there can be never exist a continuous function g defined on R such that g(x) is rational when x  is irrational and g(x) is irrational when x is rational. This chapter on Volterra also mentions about Hankel’s taxonomy of functions which was very promising, but in the end was fallen wayside.

Rene Baire  ( 1874 – 1932 )

The concept of no-where dense sets was introduced by Rene Baire . The reason behind thinking about dense and nowhere-dense sets was to classify the functions in various categories based on their discontinuity/integrability/differentiability behaviour. Many mathematicians and Scientists were after this goal of classifying functions.

Baire strongly believed that functions and sets are closely related. He developed the concepts behind nowhere-dense sets to show the vast difference between Category 1 Baire Set and Category 2 Baire Set. He also introduced topological view of the sets which had important consequences for mathematical analysis. Baire’s category theorem stated that one cannot obtain all points of a continuous interval by means of a denumerable infinity of nowhere dense sets. A differentiable function can be discontinuous. “How discontinuous can a derivative really be ?” Thanks to Baire, the answer is “Not very, for it must be continuous on a dense set”. Henri Lebesgue  ( 1874 – 1941 )

It was obvious by this time that there were problems with Riemann Integral. Both fundamental theorem of calculus and the interchange of limits and integrals were false with out assumptions that seemed overly restrictive. For example Dirichlet’s function could not be used while interchanging limits and integrals. Till Lebesgue entered the scene, mathematicians world over were trying to complete the following statement.

A bounded function f is Riemann integrable on [a,b] if and only if Df, the set of points of continuity is ???

It was evident that this missing condition was not “finite”, nor “denumerable”, not “first category”. Lebesgue was to change all that and provide a conclusive answer and complete the above statement. He developed on the concept of outer cover, developed by Axel Harnack, introduced measure of a set and thus was able to formulate an alternate way to look at Baire’s category 1 sets and category 2 sets. With his measure theoretical approach, he completed the above statement with the a simple answer

"For a bounded function f to be Riemann integrable on [a,b], it is necessary and sufficient that the set of its points of discontinuity be of measure zero”

Lebesgue measure provided a new dichotomy between small and large sets. He went on to introduce measurable functions which encompassed a much larger class of functions. Subsequently he used measurable functions to develop Lebesgue integral which was far more generic than Riemann integral. For instance Dirichlet’s function could be easily be integrated using Lebesgue integration. A whole host of things became tractable once viewed through the lens of measurable functions and lebesgue integral. Pointwise limit of a sequence of measurable function was also a measurable function. Limits and summation could be interchanged with mild conditions.Also for the fundamental theorem of calculus to hold good, there was no need to attaché restrictive conditions to the derivative. By 1904, Lebesgue theory provided a new direction to calculus and real analysis.

Sadly the book ends with Lebesgue measure. Undoubtedly there have been several developments in the field since the last 100 years which probably would fill up another book’s worth content. I wish the author writes a sequel to this book focusing on the subsequent developments.

If you want to know the major ideas developed in Calculus and Real analysis over a period of 3 centuries, this is one of the best books out there. Though this book is in theorem /proof + story format, all the proofs are explained intuitively with minimal amount of math. A book that explains the intuition behind lebesgue theory and traces the developments leading to lebesgue theory is priceless , for an aspiring quant.

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 19th 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 19th 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.

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. What’s the connection between probability and real analysis theory ? Why does one need to know at least the basics of Real Analysis to understand probability.Well, that’s because axiomatic probability is built , ground up, using the concepts from real analysis. Though you can pick up any graduate level probability book and go through it by having an intuitive sense of real analysis, there is no substitute to seeing axiomatic probability through the eyes of person who understands real analysis well.

My objective of going through this book was simple : I do not want to prove theorems, even though this book is probably meant to understand proofs better.  All I looking for is to  have  more than an intuitive sense of Real Analysis , but at the same time not to get really deep in to the subject. In that sense, I found this book to be PERFECT.

The book is organized in 8 delightful chapters, taking the reader from Real Line all the way up to Metric spaces. Let me attempt to summarize this book in words.

Chapter 1  : The Real Numbers

This chapter introduces the preliminary tools and concepts of real analysis. It starts off with an example of computing square root of 2 and shows that set of rational numbers Q do not contain such irrational numbers. To prove that square root of 2 exists with out assuming anything is impossible and hence axiom of completeness is introduced which helps us get a grip on such numbers. Axiom of Completeness abbreviated as AoC is the key axiom , based on which further concepts and tools are explored such as nested interval theorem, Cardinality of Sets, Countable and Uncountable sets, Cantor’s theorem. Epilogue mentions about continuum hypothesis, which basically states that there is no set A which follows the inequality : Cardinality of N < Cardinality of A < Cardinality of R.

Chapter 2 : Sequence and Series
Infinite series are funny creatures. They cannot be added multiplied subtracted as one does with the finite series. A beautiful introduction to the concept of convergence and divergence of series is given in this chapter. I have not seen a more lucid explanation of convergence in any other book. The illustration showing the epsilon neighbourhood of a function will forever be remember by any reader.

The following logical structure summarizes chapter 2 AoC acts as the crucial step in the proof of the Nested Interval Property (NIP). AoC is also the central step in the Monotone Convergence Theorem(MCT), and NIP is central to proving the Bolzano–Weierstrass Theorem(BW). Finally, one needs BW to prove the Cauchy Criterion (CC) for convergent sequences.One point that cannot be missed from this chapter is that MCT, CC are basically useful tools / concepts which conveniently avoid the term”limit” but still manage to talk about convergence of a series.

Chapter 3 : Basic Topology of R
The chapter begins with a fantastic introduction to Cantor Sets. Clear definitions for Open Sets / Closed Sets / Compact Sets / Open Covers / Perfect Sets / Connected Sets / Nowhere Dense sets are given with appropriate examples. Baire’s theorem is introduced in the concluding part of the chapter.

Chapter 4 : Functional Limits and Continuity

The chapter starts off with 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. These two functions are good enough to dispel the notion that functions are smooth curved graphs , an image which suits our intuition. However math / real analysis is a subject that is filled with so many examples that stump our intuition and make us go back to understanding the subject rigorously. Epsilon –Delta definition is introduced in this chapter as a method to rigorously verify the existence/non existence of a limit.

Chapter 5 : The Derivative

There are some extremely important questions at the beginning of the chapter which give a flavour of things covered . Some of these questions are as follows: Do all continuous functions have derivatives? If not how nondifferentiable can a continuous function be ? Are all differentiable functions continuous ? If a function f has a derivative at every point in a specific domain, can we say that f’ is continuous ? Can one describe the set of all derivatives ?

Mean value theorem, Rolle’s theorem , L’Hospital’s rule, Darboux’s theorem are the main aspects touched upon in this chapter. One obvious takeaway from this chapter is that differential functions are an exception than a rule. Continuity is a strictly weaker notion than differentiability

Chapter 6 : Sequences and Series of Functions

Polynomial functions are great friends to any applied mathematicians as they are continuous, infinitely differentiable and defined on all of R. They are easy to evaluate and easy to manipulate, both from the  view of algebra (adding, multiplying, factoring) and calculus (integrating, differentiating). It should be no surprise, then, that even in the earliest stages of the development of calculus, mathematicians experimented with the idea of extending the notion of polynomials to functions that are essentially polynomials of infinite degree. Such objects are called power series

One of the first things that comes to mind is “Point wise convergence” where the sequence of functions, i.e, a sequence of real numbers converge point wise to a specific function. But this isn’t enough as a few examples mentioned in the chapter show that, a sequence of continuous functions converge to a function which is not continuous. Also it is possible that a set of sequence of differentiable functions can converge to a function which is not differentiable. Thus a stronger notion of convergence is needed when one uses the limit function as a substitute for the sequence of functions. The concept of “Uniform Convergence” is introduced here which ensures that limit process for a sequence of functions retains the properties of continuity and differentiability of the originating sequence. A few tests for Uniform convergence of a series are stated, notably being Weierstrass M-Test

The all powerful series discovered by Brook Taylor is then introduced at the end of chapter .Taylor series gives a beautiful way to approximate a function to an infinite polynomial series. Lagrange’s remainder theorem is subsequently mentioned which gives an idea about the strength of Taylor series approximation for the function. The chapter then raises the question about the convergence of Taylor series to the function itself. How can we be sure that the Taylor series converges to the original function? Epilogue for the chapter tries to answer the question by analyzing the boundedness of the nth derivative of the generating function.

Chapter 7 : The Riemann Integral

Historically, integration was viewed as the inverse of differentiation. Integral of f was understood to be a function F that satisfies the relation that F’ = f . However this gives one a limited number of functions that can be integrated. What if the function is discontinuous? Thus “Why Riemann” is addressed at the very beginning of the chapter. Instead of using this inverse relationship approach, Cauchy and later Riemann introduced a fresh way to look at things by coming up with “area under the curve” approach. Partitioning the area under the curve and calling it as an integral was a breakthrough in disconnecting the inverse relationship thinking. Splicing the area under the curve to thin slices and adding up the area obtained by the splices, one inevitably stumbles on the continuity definition. Is continuity important for the Riemann sums to converge? Can it handle a function which has countable discontinuities? Can it handle a function which has uncountable discontinuities? These questions make the reader curious to move ahead in the chapter.

The chapter goes on to define Riemann integral in terms of lower sum of partitions and upper sum of partitions of the area under the curve. How does one check whether a function is integrable or not? This aspect is addressed again using the upper sum and lower sum of partitions. Thus the Riemann integral is very well explained for continuous functions, bounded functions in the initial part of the chapter. Cauchy dealt mostly with continuous/ bounded functions and thus the partition approach worked fine. However one immediately starts thinking about functions which are not continuous. The chapter then states that Riemann integral works well for countable discontinuities but could fails for some uncountable discontinuous functions (Dirichlet’s function).
What is the criterion for Riemann integrability? This question urges the curious reader to move ahead in the chapter.

Instead of talking about Lebesgue theorem and addressing the criterion for Riemann integrability, the chapter takes a detour to understand a specific aspect of integration applied to sequence of functions. When does a mathematical manipulation such as integration respect the limiting process? This question is precisely addressed in this chapter. It says that if the sequence of functions is uniformly continuous then the one can switch the limit and integral sign.

Lebesque theorem is finally stated to connect the ideas of Riemann integral and continuity. The theorem basically states that “ Let f be a bounded function defined on the interval [a, b]. Then, f is Riemann-integrable if and only if the set of points where f is not continuous has measure zero.” If one goes over the statement carefully, all the confusion relating to the nature of functions which are integrable under Riemann procedure is put to rest. Basically a function with uncountable discontinuity is also Riemann integrable if the measure of those discontinuous sets is 0. The chapter concludes with an interesting example of a differentiable function which is not Riemann integrable specifically because it fails to adhere to Lebesgue theorem.

One inevitable conclusion of this chapter is that one cannot let integrability depend on the continuous nature of functions as there are a ton of derivatives which are not continuous. Hence the logical conclusion for any reader to move to the Lebesque space. Lebesque integral encompasses far more functions than what a Riemann integral can , and the good thing about it is that whenever Riemann integral exists, it converges to Lebesque integral. Example of integrating a Dirichlet function clearly shows the strength of Lebesque integral as compared to the Riemann integral.

This book is extremely well written , a feature generally not found for most of the math books. One’s understanding of the Real line, Functions that map on to Real Line would vastly be improved by going over carefully the arguments in various chapters. This book is undoubtedly the most accessible introduction to the world of “Real Analysis”.