Self-Study Analysis: A Proof-to-Manifolds Roadmap
Direct answer: To self-study real analysis from a calculus background, follow this sequence: complete a proof-writing book (Velleman’s How to Prove It or the free Book of Proof), then work through Bloch’s The Real Numbers and Real Analysis for single-variable analysis, then Hubbard and Hubbard’s Vector Calculus, Linear Algebra, and Differential Forms for multivariable analysis and linear algebra together, and optionally supplement differential forms with geometric algebra texts by Alan Macdonald.
Table of Contents
Key Takeaways
- Single-variable calculus, including some exposure to epsilon-delta arguments, is the only prerequisite needed before starting this analysis road map.
- Velleman’s How to Prove It: A Structured Approach and the free online text The Book of Proof by Richard Hammack both cover logic, set theory, proof techniques, relations, functions, and infinite sets.
- Bloch’s The Real Numbers and Real Analysis includes a rigorous construction of the natural numbers, integers, rationals, and reals, plus proofs that pi and e are irrational.
- Hubbard and Hubbard’s Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach teaches multivariable analysis and linear algebra together and is written for readers with no prior multivariable calculus.
- Later editions of the Hubbard and Hubbard text are recommended over earlier ones, and the appendix contains proofs of the book’s major theorems.
- Alan Macdonald’s Linear and Geometric Algebra and its companion volume Vector and Geometric Calculus are suggested as optional follow-up reading on differential forms and geometric algebra.
Why Avoid Jumping Straight to Abstract Topology?
This road map deliberately delays abstract concepts rather than rushing toward them. The author states that while some students move from calculus directly into topological spaces, a more grounded, longer path makes later abstract material easier to understand intuitively. This is a matter of pedagogical preference rather than a claim about what is strictly necessary, and readers who prefer a faster route to abstraction may choose differently.
What Background Do You Need Before Starting Analysis?
Calculus and Epsilon-Delta Exposure
Single-variable calculus is the only hard prerequisite for this road map. No additional coursework beyond single-variable calculus is assumed.
Some prior exposure to epsilon-delta formulations, even a small amount, is expected from that calculus course. The more comfortable a reader already is with this technique, the easier the material that follows will be.
Why Start With a Proof-Writing Book?
A proof book teaches the vocabulary and grammar of mathematical proofs before a reader attempts full analysis texts. Proof books are described here as “a necessary evil”: they do not teach fluent proof-writing, but they do teach the symbols, logic, set theory, and standard proof techniques needed to read proofs written by others.
After finishing a proof book, a reader should be able to follow many different styles of proof and recognize common techniques, but should not expect to write proofs fluently yet. That skill develops later, through practice with analysis itself.
Recommended Proof Books
| Book | Author | Cost | Topics Covered |
|---|---|---|---|
| How to Prove It: A Structured Approach | Daniel J. Velleman | Paid | Basic logic, basic set theory, proof techniques, relations and functions, infinite sets |
| The Book of Proof | Richard Hammack | Free | Same core topics as Velleman’s text |
How Should You Read an Analysis Textbook?
Analysis books require active reading rather than casual reading. Progress through an analysis text is naturally slow: some days a reader may cover only a single page, and this slow pace is normal rather than a sign of failure. The pace improves with practice over time.
Which Book Should You Use for Single-Variable Real Analysis?
Ethan Bloch’s The Real Numbers and Real Analysis is recommended as the first analysis textbook. It is described as very rigorous while still containing substantial intuition and historical context, unlike some other rigorous texts.
Standard Topics Covered in Bloch
- Axioms for the real numbers
- Construction of the real numbers
- Limits
- Continuity and uniform continuity
- Differentiation
- Riemann integration
- Sequences and series
- Series of functions
Notable Content Not Found in Most Other Analysis Books
Bloch’s text covers ordinary calculus topics but proves every result rigorously, leaving nothing unproven. Several inclusions are singled out as unusual for an analysis textbook:
- A rigorous construction of the natural numbers (N), integers (Z), rationals (Q), and real numbers (R)
- The 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, with a proof that the Riemann integral actually measures area, and a similar treatment for the lengths of curves
- A complete characterization of Riemann integrable functions, known as Lebesgue’s theorem
- A proof that pi and e are irrational numbers
- Construction of a function that is continuous everywhere but differentiable nowhere
Walter Rudin’s analysis textbook is mentioned as a commonly cited alternative but is described here as poorly written in places, particularly its multivariable and Lebesgue integration sections, and as generally lacking intuition. The author states a preference against recommending Rudin to any reader.
How Do Multivariable Analysis and Linear Algebra Fit Together?
Multivariable analysis depends heavily on linear algebra, so a working knowledge of linear algebra is necessary before tackling most multivariable analysis results. Hubbard and Hubbard’s Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach teaches both subjects together in a single text and is suitable for readers with no prior multivariable calculus course.
Later editions of this book are recommended over earlier ones. Most of the proofs for the book’s major theorems appear in an appendix, and readers studying analysis specifically are advised not to skip that appendix. Some familiarity with basic physics is helpful but not required.
Topics Covered in Hubbard and Hubbard
- 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 and 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 to divergence, gradient, and curl
- The general Stokes theorem
Why This Book Is Recommended Over Alternatives
Several features distinguish Hubbard and Hubbard from other multivariable analysis texts, according to the author’s assessment:
- It integrates analysis and linear algebra in one text, useful even for readers who already know linear algebra
- It covers nonstandard topics such as the central limit theorem, basic differential geometry, and electromagnetism
- Its explanations of differential forms and their operations are described as more thoroughly motivated than in books such as Rudin’s
- Its nonstandard definition of the exterior derivative is presented first because it is easier to grasp, with the usual definition proven later
An Alternative: Spivak’s Calculus on Manifolds
Michael Spivak’s Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus is offered as an alternative or supplement. Its problems are considered better than those in Hubbard and Hubbard, though its exposition is considered weaker. Using both books together is suggested as a viable option.
How Can You Deepen Your Understanding of Differential Forms?
Differential forms are described as extremely important in mathematics yet frequently neglected in undergraduate curricula, and many students find them difficult on first encounter. A full understanding of differential forms is said to benefit from additional study of Clifford algebras, also known as geometric algebra, and of infinitesimal calculus, beyond what Hubbard and Hubbard alone provides.
H. Jerome Keisler’s free calculus textbook is recommended for the basics of infinitesimal calculus. Alan Macdonald’s Linear and Geometric Algebra is recommended for geometric algebra; readers with strong linear algebra backgrounds can skip much of it and focus on Part II. This book also covers quaternions and explains what physicists mean by a pseudovector.
Readers who enjoy the geometric algebra approach can continue with Macdonald’s Vector and Geometric Calculus, which treats multivariable analysis using the language of geometric algebra. This follow-up text is described as optional.
What Comes After This Road Map?
Completing this sequence rigorizes the calculus a reader already knows, but analysis as a field extends far beyond rigorous calculus. Further posts in this series are planned to cover what comes next in the study of analysis.
Frequently Asked Questions
Do I need more than calculus to start this analysis road map?
No additional coursework is required beyond single-variable calculus. Some prior exposure to epsilon-delta formulations from that calculus course is assumed, though it does not need to be extensive.
Should I start with Rudin’s Principles of Mathematical Analysis?
This road map does not recommend Rudin’s textbook. It is described as poorly written in places, particularly the multivariable and Lebesgue integration sections, and as lacking intuitive explanation compared to Bloch’s text.
What is the free alternative to Velleman’s proof book?
Richard Hammack’s The Book of Proof, available freely online, covers the same core topics as Velleman’s How to Prove It: basic logic, set theory, proof techniques, relations and functions, and infinite sets.
Which edition of Hubbard and Hubbard should I buy?
Later editions of Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach are recommended over earlier ones, though the specific edition number is not given here.
Is studying geometric algebra required to understand differential forms?
It is presented as beneficial rather than strictly required. Alan Macdonald’s Linear and Geometric Algebra and Vector and Geometric Calculus are both described as optional supplements for deepening understanding of differential forms.
How fast should I expect to progress through an analysis textbook?
Progress is expected to be slow, sometimes only a single page per day of active reading. This pace is described as normal rather than a sign that a reader is struggling, and speed is said to improve with practice over time.
Advanced education and experience with mathematics
As a followup to my previous post, a while back I reached out to Ethan Bloch about his book and he had this to say (super helpful guy btw):> "As for Dedekind Cuts, that is indeed a very heavy duty topic, and is more > tedious and technical than a lot of other parts of the book. In truth, when > I teach our standard real analysis course, I start with Chapter 2, skipping > the Dedekind Cuts entirely. The only reason to do Dedekind Cuts is if > someone has a particular interest in that subject, or if someone has > already seen some real analysis, or the like. The reason the Dedekind Cuts > are at the start of the book is because logically that is where they > belong, though pedagogically it isn’t the most user-friendly place to start."He also suggested that students who find proofs challenging at first will often benefit from doing an intro abstract algebra course prior to diving into real analysis. As luck would have it [USER=205308]@micromass[/USER] has an excellent guide for that: https://www.physicsforums.com/insights/self-study-algebra-part-ii-abstract-algebra
For anyone like me who is finding Bloch to be a tough first book on analysis, I'd like to suggest "Understanding Analysis" by Abbott as a supplement. So far, I'm finding it to be more approachable, especially since I'm not able to dedicate time every day to work through it.https://www.amazon.com/Understanding-Analysis-Stephen-Abbott/dp/1493927116
What got me into analysis started with the concept of "sigma algebras". Once I grokked that, it was all clear sailing.
is there at least one more or less famous mathematician in 20 century who had self- studied analysis sitting at his home and did not graduate university?
Thanks for taking your time to write a reply, I appreciate that! I already knew of Jänich's great text but not of Gamelin and Greene's book. I'll be sure to check out this one too.
Another book which is excellent for learning general topology is Gamelin and Greene's Introduction to Topology 2e. It has excellent exercises, is slim with few wasted words, and at the same time manages to not skimp on needed explanation at some of the common sticking points in learning topology. It would be a good companion to Lee's books on manifolds.Also, Klaus Jänich's Topology is an excellent supplement to any path of learning topology. Jänich doesn't have a full complement of exercises, and doesn't always have the precise pedagogy other texts have. However, the book is full of excellent intuitive explanations and diagrams.Learning topology well will help a lot with your later explorations of analysis, as well. Even many proofs in basic real analysis are more elegant and easier to understand when phrased in topological terms rather than in epsilon-delta form.
Thanks for your help! I initially thought Lee's book already required point-set topology but that was Introduction to differentiable manifolds. I'll take a look at this book in the university library and I'll be sure to PM you if I need more information. Thanks again!
In your case, it seems that Lee's "Introduction to topological manifolds is ideal". Here's why1) The prerequisites needed are a good experience with set theory proofs and metric spaces. You seem to have this, so you meet all the prereqs. I do advise going through the appendix first.2) Despite it saying "graduate texts in mathematics", this is actually one of the more intuitive and easier texts on the subject. I personally think it's perfect for a first encounter. You might want to go through another book later though, since it doesn't cover everything you need to know.3) It is especially made for somebody interested in differential geometry and it focuses a lot on manifolds.https://www.amazon.com/Introduction-Topological-Manifolds-Graduate-Mathematics/dp/1441979395Feel free to PM me if you want more help.
Hello micromass and thank your for this great Insights!I have some familiarity with Real Analysis from Abbott but the problems were a bit tough for me at the time. I'm going through Tao's Analysis books with a friend of mine right now and he's giving me additional exercises since he already covered the subject.My question is: I'm currently self-learning Algebra (with Artin and Pinter) and Analysis, do you think I have the proper prerequisites to start learning General Topology? I have the whole summer to work on mathematics considering I'll start university (as a math major this time) in September. It is not exactly my first encounter with topology, but I never covered compactness or connectedness for instance. I have some familiarity with metric spaces.What book would you recommend considering my main interest lies in differential geometry and mathematical physics? Most differential topology books I know assume a course in point-set topology.Thanks for taking your time to help me!
“dang it, I have signed myself up for a self study on analysis. I just got drawn into slowly but surely by reading the insights and looking at the recommended texts.
I need to get material on the language and use of sets, for some reason sets were a big thing in high school but by the time I got into first year high school they had been discarded as a thing to teach students.
any theories why educators felt sets were so important I think until the 70’s then fell off the face of the high school curriculum by the late 70’s/early 80’s.
so ya, I am not familiar with the language and from my scan of the insight, analysis is mainly written in the language of sets??”
Yes, I’m afraid the notions of sets are absolutely crucial to everything mathematical. I recommend Velleman’s “how to prove it” to get acquainted with sets. Although any proof book will contain enough material on it.
dang it, I have signed myself up for a self study on analysis. I just got drawn into slowly but surely by reading the insights and looking at the recommended texts.
I need to get material on the language and use of sets, for some reason sets were a big thing in high school but by the time I got into first year high school they had been discarded as a thing to teach students.
any theories why educators felt sets were so important I think until the 70’s then fell off the face of the high school curriculum by the late 70’s/early 80’s.
so ya, I am not familiar with the language and from my scan of the insight, analysis is mainly written in the language of sets??
”
1) What are the most important theorems that one must remember and master in analysis, I mean which theorems will be used the most in later courses like functional analysis or differential geometry?
”
Everything. I’m sorry, but that’s the way it is. Single variable calculus is so immensely important, every theorem you encounter is something you should deeply understand and know. I can’t say anything is less important than something else, because that would be wrong.
Most important are the techniques though. Making an epsilon-delta proof. Proving a sequence exist and converges. Proving a continuous function with one positive value has an entire open interval of positive values. Etc. Stuff like that are stuff you are expected to do very well. That you forgot a theorem is not so bad, you can always look it back up. But you should be able to handle these techniques cold.
”
2) I am currently self studying analysis using two different books, Intro to RA by Bartle and Sherbert 3rd ed. and Understanding Analysis by Abbot, how do you see these books, and do you recommend me to solve all the problems in these books? if not, which problems shall I do ?
”
Yes, you should solve all problems. Real analysis is so fundamentally important to later courses that you should take all the practice you can get. Like I said, the techniques are most important, and you only learn them by doing problems. Bartle is a really nice book and Abbott is cool too. I enjoy all of Bartle’s books very much. You won’t got wrong with them.
Don’t take this intro analysis lightly. I know for most people this isn’t really fun. It’s just all calculus, but with annoying proofs. But spend as much time as you need on this stage. Don’t rush it. You don’t want a bad foundation in this kind of analysis! Every kind of analysis (functional analysis, complex analysis, global analysis) depends on knowing this very very well.
For multivariable calculus, things change though. The differentiation part is very important: partial and complete derivatives, implicit and inverse function theorems, etc. The integration part is far less important since Lebesgue integrals generalize it much more neatly. In the end, you’ll use the Lebesgue integral everywhere and you will never care about the Riemann integral anymore. Differential forms on the other hand, are crucial, even though they are very underappreciated in the undergrad curriculum (which I think is a really awful mistake).
“Gotcha. I will definitely have to become more comfortable with linear algebra. The only thing I’ve done apart from a standard undergraduate was in my digital signal processing course where we learned about Minkowski spaces. Our first HW assignment had me stumped on the following problem:
For vector space [itex]l^p(mathbb{Z})[/itex], show for any [itex]p in [1,infty)[/itex] the vectors in [itex]mathbb{C(mathbb{Z})}[/itex] with finite [itex]l^p(mathbb{Z})[/itex] norm form a vector space.
He had talk about Minkowski’s inequality during the first lecture and I didn’t even think to use it! o:)
Thank you for responding and I will get to work right away. =)”
Yeah, those are standard first problems. The Minkowski inequality is proven in Kreyszig. Another book which isn’t really functional analysis but contains a lot of relations with the subject is Carothers real analysis book. It’s very well written.
“I will post about functional analysis soon. But if you’re comfortable with single-variable analysis (mainly continuity and epsilon-delta stuff) and very comfortable with linear algebra (the more the better, but definitely abstract vector spaces, linear maps, diagonalization, spectral theorem of symmetric matrices, dual spaces), then you can start functional analysis. A very very good book is Kreyszig’s functional analysis book. Any other functional analysis book requires quite a bit more analysis including measure theory. But I would start with Kreyszig and move to a more advanced book later.”
Gotcha. I will definitely have to become more comfortable with linear algebra. The only thing I’ve done apart from a standard undergraduate course was in my digital signal processing course where we learned about Minkowski spaces. Our first HW assignment had me stumped on the following problem:
For vector space [itex]l^p(mathbb{Z})[/itex], show for any [itex]p in [1,infty)[/itex] the vectors in [itex]mathbb{C(mathbb{Z})}[/itex] with finite [itex]l^p(mathbb{Z})[/itex] norm form a vector space.
He had talked about Minkowski’s inequality during the first lecture and I didn’t even think to use it! o:)
Thank you for responding and I will get to work right away. =)
“Hi Micro! Thank you for the advice!
Where would you say functional analysis fits? I don’t believe I am ready right now, but I’d like to know in what direction I should be going after completing single-variable analysis.”
I will post about functional analysis soon. But if you’re comfortable with single-variable analysis (mainly continuity and epsilon-delta stuff) and very comfortable with linear algebra (the more the better, but definitely abstract vector spaces, linear maps, diagonalization, spectral theorem of symmetric matrices, dual spaces), then you can start functional analysis. A very very good book is Kreyszig’s functional analysis book. Any other functional analysis book requires quite a bit more analysis including measure theory. But I would start with Kreyszig and move to a more advanced book later.
“Perhaps I am confusing them a bit but I still need to improve the mathematical side.”
So it is officially 5000 1 in confusions; you have already cleared up around 5000 of mine in the Computers forum to this one of yours ;).
“I wonder if you are mixing data analytics with Mathematical Analysis? Hope I am not saying something dumb. And interesting that in Spanish, TODO, without spaces means everything. Hope your list does not include _everything_ and that your load is lighter than that :).”
Perhaps I am confusing them a bit but I still need to improve the mathematical side.
“Thanks micromass. I have a large, related TODO list which is why I haven’t gotten around to asking about the analytics yet. I think that it’s going to be another six months before I’ll have the time to start. This is where I plan to put it to work though – [URL]https://d3js.org/[/URL]”
I wonder if you are mixing data analytics with Mathematical Analysis? Hope I am not saying something dumb. And interesting that in Spanish, TODO, without spaces means everything. Hope your list does not include _everything_ and that your load is lighter than that :).
“Feel free to PM me for any further information! Or post something in this thread.”
Thanks micromass. I have a large, related TODO list which is why I haven’t gotten around to asking about the analytics yet. I think that it’s going to be another six months before I’ll have the time to start. This is where I plan to put it to work though – [URL]https://d3js.org/[/URL]
“Thanks for this micromass. I’ve been meaning to post a question asking for exactly this. :smile:”
Feel free to PM me for any further information! Or post something in this thread.
Thanks for this micromass. I’ve been meaning to post a question asking for exactly this. :smile:
Nice information thanks for sharing
Thank you for your post, very helpful, looking forward to the next part, but I wish that there was more elaboration on why one should study real analysis, and the importance of real analysis on later courses of mathematics, also more elaboration on the struggle of self learner and how to overcome them, with examples from your experience, since you have studied many courses by your self.I have some questions for you,1) What are the most important theorems that one must remember and master in analysis, I mean which theorems will be used the most in later courses like functional analysis or differential geometry?2) I am currently self studying analysis using two different books, Intro to RA by Bartle and Sherbert 3rd ed. and Understanding Analysis by Abbot, how do you see these books, and do you recommend me to solve all the problems in these books? if not, which problems shall I do ?Thank you again dear Micromass for your very helpful contributions to the forums.
Hi Micro! Thank you for the advice!Where would you say functional analysis fits? I don't believe I am ready right now, but I'd like to know in what direction I should be going after completing single-variable analysis.
Thank you again Micromass ! I'm not really sure how much time it will take me to complete your guides but I'll try to put as much efforts as needed. Very helpful!
Thanks MM! Looking forward to the next in the series :)