Self-Study Analysis Roadmap: From Proofs to Manifolds

Introduction

This is a sequel to my posts on self-studying mathematics. I have already given a very detailed road map on how to study high school mathematics and calculus. In this post and the next ones, I will try to give a very detailed road map on how to self-study analysis to reach a high level.

Avoiding premature abstraction

In planning this road map, I have chosen a very deep and grounded path. In my road map, you will not reach abstract concepts quickly. I have chosen a path that makes the abstract and advanced concepts you will meet later on much more intuitive and approachable.

So, while it is possible to study topological spaces fresh out of calculus (and I know people who have done this), I avoid such approaches in favor of a more long-winded but more grounded path.

Prerequisites

Required background

First, I will try to detail how to get started in analysis. The prerequisites you need to have should be clear: single-variable calculus is a necessary condition. I will not assume anything more than single-variable calculus.

Epsilon-delta experience

I do assume that in single-variable calculus you spent some time, which does not need to be much, on epsilon-delta formulations. The more experience you have with this crucial tool, the easier you will find the material below.

Proofs: logic and techniques

Why learn proofs?

First, it is important to be well acquainted with proofs. For this, you will need to go through a proofs book. I’m not a fan of proof books; I consider them a necessary evil. When doing proof books, you will not learn how to write proofs down fluently. You will, however, learn the basic vocabulary and grammar of proofs.

You will learn the language in which proofs are written (logic and set theory), how to construct elementary proofs in set theory and logic, and the major proof techniques.

After you have completed a proof book, you will be able to read many different proofs more easily, and you will recognize the different symbols and techniques. Do not expect to be able to write proofs fluently; that is not the purpose of a proof book.

Recommended resources

As a recommendation, I give Velleman’s How to Prove It: A Structured Approach http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995.

Topics covered

You will learn the following topics:

• Basic logic
• Basic set theory
• Proof techniques
• Relations and functions
• Infinite sets

A nice (and free!) alternative is The Book of Proof, available here http://www.people.vcu.edu/~rhammack/BookOfProof/. This covers the same topics.

Single-Variable Real Analysis

Getting started with analysis

Once you have become somewhat proficient at handling proofs, it is time to dive into analysis. Analysis is quite a difficult topic for most newcomers, but rest assured that you’ll do fine given enough practice and thinking. Don’t underestimate an analysis book.

How to read analysis books

An analysis book (like most math books) is not something you can read before sleep. It is something you must read actively. If you read it actively, you will go very, very slow. Don’t think you’ll do more than several pages a day; sometimes you won’t even get further than one page. This might be discouraging at first, but it gets better over time.

Recommended first book

As the first analysis book, I recommend Bloch’s The Real Numbers and Real Analysis. This is a very rigorous book, but it contains a lot of goodies. The standard topics you will encounter in Bloch are:

Topics in Bloch

• Axioms for real numbers
• Construction of the real numbers
• Limits
• Continuity and uniform continuity
• Differentiation
• Riemann integration
• Sequences and series
• Series of functions

Notable contents in Bloch

This might look like an ordinary course in calculus, and indeed the topics covered in Bloch tend to be those of an ordinary calculus course, except that everything is proven rigorously. Nothing is left unproven. Some nice goodies that can be found in Bloch and not in most other analysis books:

• Rigorous definition and construction of N, Z, Q, and R.
• Recursion theorem
• A rigorous definition of decimal expansions
• Several equivalent forms of completeness (including a proof that completeness of R is equivalent to the intermediate value theorem)
• A rigorous definition of area and a proof that the Riemann integral does measure the area. Same for lengths of curves.
• A complete characterization of Riemann integrable functions (Lebesgue’s theorem)
• A proof that pi and e are irrational
• Construction of a continuous, but nowhere differentiable function
• Much intuition and historical notes

An often-cited alternative to learning analysis is Rudin. I consider that book to be poorly written in places (especially the multivariable part and the Lebesgue integration part). Rudin also tends to ignore any kind of intuition. I would not recommend Rudin to anybody.

Multivariable Analysis and Linear Algebra

Link with linear algebra

After we have rigorized single-variable calculus, it is time to rigorize multivariable analysis. Multivariable analysis is deeply tied to linear algebra, so you will have to know linear algebra to understand most multivariable analysis results.

Recommended integrated text

Luckily, there is an excellent book that teaches both linear algebra analysis. The book is perfectly suitable for somebody who has not taken a multivariable calculus class before. The book in question is Hubbard and Hubbard, Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. It can be found here: http://matrixed.com/5thUnifiedApproach.html. Note that the later editions are significantly better than the earlier ones, so try to get a later edition if possible. Also note that the proofs of most of the big theorems are in the appendix; if you’re learning analysis, then this appendix is not something you should ignore.

Some acquaintance with basic physics is helpful, but not necessary.

Topics you’ll learn

You will learn

• Vectors in R^n
• Matrices and matrix computations
• Limits and continuity in R^n
• Differentiation in multiple dimensions
• Vector spaces
• Linear transformations
• Eigenvectors and eigenvalues
• Newton’s algorithm
• Implicit and inverse function theorems
• Manifolds and tangent spaces
• Taylor polynomials in R^n
• Finding maxima/minima, and Lagrange multipliers
• Determinants
• Integration in multiple dimensions
• Fubini’s theorem
• Change of variables theorem
• Introduction to Lebesgue integrals
• Curvature of manifolds
• Differential forms on manifolds
• Exterior derivatives and their relation with div, grad, and curl
• The general Stokes theorem

Why I recommend it

Some reasons why I like this book over any other book:

• An integrated approach to both analysis and linear algebra (note that even if you know linear algebra already, this book is still good!)
• Many cool nonstandard topics like the central limit theorem, basic differential geometry, and electromagnetism.
• Very intuitive explanations of differential forms and their operations. They are motivated thoroughly unlike other books (Rudin).
• A very nonstandard definition of the exterior derivative makes it much easier to grasp its meaning. The usual definition is proven later.

Alternative: Spivak

Another nice book you might want to look at is Spivak’s Calculus on Manifolds http://www.amazon.com/Calculus-Manifolds-Approach-Classical-Theorems/dp/0805390219/. It contains a lot of good problems (better problems than Hubbard), but the exposition of Hubbard is a lot better. Using both Spivak and Hubbard might be a nice idea.

More on differential forms

Filling gaps

Differential forms are extremely important in mathematics and are very often ignored in undergraduate mathematics. Many students find them troubling at first. While Hubbard gives a very nice explanation of them, I feel that an understanding of differential forms would not be complete without some knowledge of Clifford algebras (also known as geometric algebra) and infinitesimal calculus.

Further reading

The basics of infinitesimal calculus can be learned from Keisler’s free book (see my self-study post about calculus). Geometric algebra is treated quite well in MacDonald’s Linear and Geometric Algebra http://www.amazon.com/Linear-Geometric-Algebra-Alan-Macdonald/dp/1453854932 (depending on your knowledge of linear algebra, you can skip a lot of this book; only part II is important here). This book will also teach cool topics like quaternions, and what physicists mean when they talk about a pseudovector.

If you like the approach of geometric algebra, you can follow this up with MacDonald’s Vector and Geometric Calculus, which treats multivariable analysis with the language of geometric algebra. This is optional though. http://www.amazon.com/Vector-Geometric-Calculus-Alan-Macdonald/dp/1480132454

Conclusion

What’s next

After reading all this material, you have rigorized all of calculus. But don’t think the analysis ends here. No way: your journey through analysis is just beginning. Analysis is way more than just rigorous calculus. In my next posts, I will tell you how to get to the good stuff.