About This Book
In my opinion, analysis is perhaps the best undergraduate course we offer students for several reasons (and I’m a topologist, so this isn’t just preference for my own field shining through!):
- Its Rigorous: Precision and clarity of thought - the foundations of mathematical proof - are fundamental skills that one learns in a mathematics degree. And, the material of real analysis naturally lends itself to practicing this material: from limits (\(\forall\epsilon\exists N\forall n>N\)…) to continuity (\(\forall\epsilon\exists\delta\forall x \textrm { s.t.} |x-a|<\delta\)…) the definitions of intuitive ideas turn out to be more subtle than one originally imagines.
- Its Familiar: Many fundamental results in real analysis show up in calculus courses or before: from knowing that the square root is continuous to using differentiation to find maxima and minima. This prior familiarity allows us to wade deeper into technical arguments (and consequently get farther into the material) than in subjects where both the concepts and proof techniques are new.
- There are Surprises: While the foundational results may be familiar, real analysis is a far cry from just filling in overly-pedantic details to known results. Indeed, the entire theory is a battle to preserve our intuition against harsh reality: from the nonexistence of infinitesimals to the wonderful variety of pathological functions there are surprises lurking just below the surface in almost every topic.
- There’s Drama: Mathematics is more than just a collection of theorems; its a field of study, developed by many humans over several millennia on Earth. This human side to mathematics is crucially important to its its practice; while the theorems we’ve proven rise above the messy contingent details of life on this planet, which theorems we’ve proven is a function of our history. And the history of analysis is rich; from the methods of the ancients to the feud of Newton and Leibniz to the struggles of formalization, important theorems and historical turning points go hand-in-hand.
- It Bridges the Pure/Applied Divide: As mathematics grows it may feel a bit fractured into “pure” and “applied” subdisciplines especially at the undergraduate level (not to mention “statistics” and “theoretical physics”; earlier branches of our common family tree). But analysis is fundamental to both sides, laying the foundations for modern Geometry / Lie Theory / Dynamical Systems, as well as Mathematical Modeling, Numerical Analysis and Optimization.
As one of the cornerstones to an undergraduate mathematics degree, there are already many excellent real analysis texts out there. So what makes it worth it to contribute yet another tome to the stockpile? The story of analysis is as broad as it is deep, and there are many narratives one can tell: I hope to tell one that emphasizes the points below.
A Sequences Forward Approach
The definition of sequence convergence is one of the first “nested quantifier definitions” to appear in analysis, and proving many theorems about sequences up front provides a fruitful playground for getting used to such definitions (and proof strategies, like the “\(\epsilon\)-\(N\) game”).
Taking advantage of all this work done early on, this book takes the sequence as the fundamental object in analysis, and develops tools to study other ‘nested quantifier type’ definitions in terms of sequences. In particular
- We prove that (the standard \(\epsilon\)-\(\delta\)) continuity of a function \(f\) at a point \(a\) is equivalent to the following: for every sequence \(x_n\to a\), the sequence \(f(x_n)\) converges to \(f(a)\). This allows for simple proofs of many facts about continuity building directly off of limits.
- We prove that the \(\epsilon-\delta\) notion of limit of a function is equivalent to the following sequence version: \(\lim_{x\to a}f(x)=L\) if for all sequences \(x_n\to a\) with \(x_n\neq a\), the sequence \(f(x_n)\) converges to \(L\). This allows one to translate sequence convergence theorems directly to facts about limits of functions, and provides a natural way to work with left and right hand limits.
- After Darboux integrability we discuss the Riemann integral, and its definition involving all sequences of shrinking partitions also allows us to prove several properties of integrals from corresponding statements about sequences.
Discovering the Elementary Functions
Some parts of real analysis can be taught completely abstractly, speaking only of functions \(f\) and \(g\), and never specifying particular functions at all. But other parts of the field are dedicated specifically to understanding and constructing specific functions, from the familiar exponentials logs and trigonometric functions to more esoteric special functions like the gamma function, bessel functions, and jacobi elliptic functions.
This book attempts to show students a bit of both sides of analysis, by building into the main text a construction of the exponential and logarithmic functions, and allowing them to work out a full construction of the trigonometric functions as a final product. We define these elementary functions via functional equations, so we call a function an exponential if it is a continuous nonconstant solution to \(E(x+y)=E(x)E(y)\) and \(L\) a logarithm if its a (continuous, nonconstant) solution to \(L(xy)=L(x)+L(y)\).
The work to understand these functions is spread out over several sections of the text: whenever we learn new material (continuity, differentiability, power series, integration) we illustrate it by making more progress on understanding the elementary functions. One of the highlights of the course is the construction of the exponential as a power series, and explorations of this power series in further mathematics.
Axiomatic Integration
In contrast to many real analysis texts, we introduce the integral axiomatically, by proposing three axioms that anything worthy of being called ‘an integral’ ought to satisfy (this approach is based on that carried out out in Serge Lang’s book, as well as in the lecture notes of Pete Clark). These axioms specify only that (1) the integral of a constant function is the area of a rectangle (2) if \(f\leq g\) on an interval then their integrals inherit the same inequality, and (3) the subdivision rule: the integral from \(a\) to \(b\) is the same as the sum of the integrals over \([a,c]\) and \([c,b]\) for \(c\in[a,b]\).
From these axioms alone, we prove that if \(\int\) is any integral satisfying these axioms and \(f\) is a continuous integrable function, then the fundamental theorem of calculus holds. From here, we can prove many things (contingent on an integral existing) in a way that does not depend on the messy details of any particular construction. Indeed, we prove for integrable continuous functions
- The integral is linear, when restricted to continuous functions.
- You can integrate power series term by term
- U-substitution and integration by parts are valid integration techniques, when restricted to continuous functions.
The rationale for this approach is twofold. One, working with any particular integral (Riemann, Darboux, Lebesgue, etc) involves complicated arguments where the spirit can be lost in the details. But working axiomatically forces an argument to rely only on simple geometric premises. Second, the existence of so many different integrals (with different advantages/disadvantages) can be rather confusing to a beginning student, the axiomatic approach clearly separates out facts that are true for any possible integral (things you can prove from the axioms) from those that are true of a particular integral (things you can only prove using a particular construction).
Of course, there certainly remains an important place for showcasing at least one construction here: namely to prove that this entire theory isn’t vacuous! But the importance is lessened, and students can treat the (sometime daunting) theory of the Darboux integral as more of a ‘covering all our bases’ than as a fundamental topic that needs to be deeply understood before moving on. To be sure, there is still some payout from the construction: we prove that the Darboux integral really is linear (on its entire space of integrable functions, not just the continuous ones), we generalize the integrability of power series term by term to a version of Dominated Convergence for the Darboux integral, and we use the ability to calculate integrals with Riemann sums to provide often-unseen infinite series converging to the natural logarithm and \(\pi\).
Historically Important Problems
We make sure not just to cover the classic theorems required in a first analysis course, but also use them to solve problems of historical significance. In particular we rigorously discuss the following:
- The babylonian approximation to the square root of 2, as an introduction to monotone sequences, recursive sequences, and later continued fractions.
- Archimedes method of measuring the circle by exhaustion as a motivation for the nested interval theorem and for developing a theory of subsequences.
- Archimedes quadrature of the parabola as an introduction to the geometric series.
- The construction of the Koch snowflake fractal.
- Euler’s formula and the relationship of the trigonometric functions to the complex exponential.
- As a final project: a solution to the Basel problem, proving \(\sum_{n\geq 1}\frac{1}{n^2}=\frac{\pi^2}{6}\) using the material from the course.
A Unified Theory of Limit Switching
To determine when it is possible to permute a limit with an infinite sum, we prove Tannery’s theorem (which we call Dominated Convergence, as it is a special case of the Lebesgue Dominated Convergence theorem applied to the measure space \(\ell^1\)). As the course continues, any time we are faced with needing to exchange some limiting process with a sum, we start from this theorem and prove a relevant generalization. In all, we collect the following:
- Dominated Convergence for series
- Dominated Convergence for limits of series of functions
- Dominated Convergence for derivatives of series
- Dominated Convergence for integrals of series.
This gives a set of easily memorable conditions (because they are the same, or very similar in all cases) on when you can pull a limit inside of one of these operations.
To keep all limit-sum-exchanges in this family of similar theorems we depart from several usual topics in a first analysis course: we do not discuss uniform convergence, nor its implications for differentiation/integration of sequences of functions. Longer term, this book will be expanded into a one year analysis course, and the entire focus of the second half will be functional analysis; where a thorough treatment of these topics will be undertaken.
A Foundation for Functional Analysis
This book is aimed to provide undergraduates a foundation that will make it possible to study some aspects of functional analysis (convergence in function spaces, the linear algebra of differential equations, fourier series, and the calculus of variations). Some choices that were made specifically to help with this transition to more advanced analysis are
- Basing all limit switching off dominated convergence theorems. This makes the transition to the Lebesgue integral natural, and makes it clear why such an upgrade is needed (as our Dominated convergence theorem for the Darboux integral is slightly weaker than our other dominated convergence theorems; and the theorem we would have expected is exactly the one which is true for Lebesgue).
- Defining things axiomatically (the real numbers via the completeness axiom, the elementary functions by functional equations, and the integral by area properties) models many of the definitions of higher mathematics, where the focus is on what an object is for versus on any specific construction.
- We spend an extended amount of time discussing the exponential function, and how to use such a definition to extend the domain of a function from its original home (the real numbers) to more abstract spaces (complex numbers, matrices, linear maps, differential operators). The brief introduction to differential equations foreshadows some of the material coming in the sequel.