
Ask HN: Resources to learn real analysis? - pedrodelfino
Hi,
I am an undergradute student of Applied Mathematics in Brazil. This semester, I will do a Real Analysis course and I am keen on learning this subject!<p>I love HackerNews. This is a great community with awesome people and marvelous content going on. It would be nice to receive some advice from you guys.<p>The professor is using the book &quot;Analysis&quot;, from Terrence Tao. I am looking forward to supplementary material that will help me absorb this and gain some intuition.<p>1 - Is there a YouTube content particularly good for this topic?<p>2 - Is there some specific good strategy to study Analysis?<p>I really like to study doing exercises and, then, checking the answer. Not just the final answer but the whole answer.<p>This is not always available. Slader is a great website for that. Maybe there is an even a better resource than Slader that I do not know.<p>Thanks in advance!
======
ghufran_syed
Without criticizing any other people’s recommendations on this thread, I think
it’s easy for people who have been doing proof-based math for a few years to
recommend books that look clear and easy to them _now_ , without remembering
the time, effort and other support (e.g. great teacher and classmates) that
may have been necessary to make use of that book a good experience.

Or maybe they are just way smart than me? :)

Either way, when considering possible books to use, I would ask the following
question - is there a chance this is “too easy” to use / read, while still
claiming to be about analysis? (I.e. calculus books fail this test because
they don’t say they are about analysis). Then start with the easiest one
unless there are really good reasons not to.

My own specific advice would be:

1) make sure you have had practice with proof based math before. If not, or
you need the practice, get a copy of chartrand’s “introduction to mathematical
proof” and do some exercises from the first 10 chapters. If you can do them
easily, move onto analysis, if not, work through those 10 chapters first.

2) The book I personally like best for self-study is Abbott, “Understanding
analysis” particularly if you can get the solutions manual, I think the
explanations of the proofs are very good.

3) I would also recommend Lara Alcock’s book “How to think about analysis”,
which is NOT a textbook, but has a lot of useful information and advice on how
to learn analysis.

Also, obvious but worth repeating, if you are taking less than one hour per
page to get through an analysis text, or don’t have pencil and paper in hand
while going through the book, “ur doing it wrong” :)

~~~
defen
On the topic of newcomers to proof-based math - "How to Prove It" is a great
resource: [https://www.amazon.com/How-Prove-Structured-
Approach-2nd/dp/...](https://www.amazon.com/How-Prove-Structured-
Approach-2nd/dp/0521675995)

~~~
jfv
This book made proof based math approachable for me. There are mechanical
aspects to proofs that are probably best practiced in a structured way and
this book does that by also acquainting you with set theory.

------
jhck
Tao's Analysis I is fantastic. There's a review available from the MAA:
[https://www.maa.org/publications/maa-
reviews/analysis-i-0](https://www.maa.org/publications/maa-
reviews/analysis-i-0)

The book doesn't touch on applications. Since you're studying applied math,
you might want to supplement it with something like "Calculus with
Applications" by Peter Lax and Maria Terrell.

On YouTube one can find lecture videos from a real analysis course given by
Francis Su (former president of the MAA):
[https://www.youtube.com/playlist?list=PL0E754696F72137EC](https://www.youtube.com/playlist?list=PL0E754696F72137EC)

~~~
tashi
Holy cow, check out that review: don't waste your time trying to teach this to
an average undergraduate math major because it "would largely amount to swine
facing pearls not meant for them."

~~~
pvg
It’s an allusion to a well-known bit in the New Testament, not actually
calling undergrads swine.

[https://en.wikipedia.org/wiki/Matthew_7:6](https://en.wikipedia.org/wiki/Matthew_7:6)

~~~
tashi
I'm familiar with the expression. Its use here indicates that the reviewer
believes the world is divided into those who are smart enough to follow the
book and those who are wastes of time.

If it's not obvious why that sucks, imagine going to this professor's office
hours if you're having trouble following a proof. Imagine that, rather than
thinking you might just not have seen this style of proof before and trying to
walk you through it, he indicates you aren't cut out to understand it and
suggests you change your major.

(If it sounds like I'm being too harsh based on that one data point, after I
read the review I went and looked up the professor on ratemyprofessor.com, and
that seems to be his approach to struggling students: that he's weeding out
people who just can't learn the material, not people who just got stuck
because they're lacking the mathematical sophistication or exposure or
confidence they need, or people who are learning more slowly and will be fine
if they put in more time and effort and come up with better strategies. Not
sure how a professor at Loyola Marymount decided he's qualified to be The
Gatekeeper of Mathematics, but there it is.)

~~~
pvg
It doesn't really imply that. He just says, a bit stridently, that he thinks
the book is targeted at people who will make maths the primary focus of their
study. The other stuff is mostly you extrapolating a lot from a single
offhanded remark. Reviews are not office hours. Going out to cherrypick
additional 'data' doesn't really make your conclusion more sound. The dude
wrote some sentence you don't like and next thing you know he's Vinz Clortho
the Keymaster of Office Hours.

~~~
tashi
Well, if you're going to take the high road and give him the benefit of the
doubt, I won't argue further. Maybe I'm just being cynical.

But in the review he wasn't just talking about "people who make maths the
primary focus of their study." If he had just been talking about students'
areas of interest or work ethic I would never have objected. It was
specifically this: "One should pick one's audience carefully... and treat
these gifted kids like apprentices." In my experience that approach misses a
lot of talented people who were different enough not to get matched by the
"gifted" filter.

~~~
aoki
most serious US math programs have honors sequences that are aimed at students
who intend to go for a PhD and who are often starting with post-calc-I/II/III
courses from the very beginning. tao says right in the introduction that this
text is for such an honors sequence at UCLA. so, yeah, this book is explicitly
written for undergrads who are being groomed to enter elite graduate programs
because they did the work (before college) to fit that profile.

in the US, i believe the real analysis course for non-honors courses at R1s is
often based on something like ross, abbott, or bartle & sherbert, whereas for
non-R1s (where most math majors will be teachers) it may be based on something
like lay or wade instead. these books are more accessible to students with
less mathematical maturity.

i think what the reviewer is saying is that it would be a mistake to use this
book in one of those non-honors courses with a poor faculty-student ratio.
even if you do have some students who have the interest and ambition, you'd be
doing a disservice to the rest. you'd be exceeding the level of interest for
most and unable to support anybody adequately.

------
yshklarov
I don't understand why so many people recommend baby Rudin (Principles of
Mathematical Analysis). The presentation in Rudin is not merely terse, but
also quite dry and unmotivated. I suggest you avoid it--regardless of how much
talent or maturity you have. There are plenty of more interesting texts which
will teach you just as much: Spivak and Pugh are nice, I also recommend the
recent two-volume work by Zorich.

By the way, as you aquire experience you'll gain confidence and get over the
urge to always check your answers. Here's a good exercise with a built-in
answer key: When reading a text, every time you get to a result (claim,
theorem, etc) try to prove it on your own before you continue. You probably
should be doing that more often than not.

In any case, don't stick to just one text/source. Shop around, read a few
pages here and there before you settle on something. There's no way a stranger
on the internet can make a good recommendation: Find what works for you. The
most important thing is that you're fully engrossed!

~~~
westoncb
I've wondered that too. My conclusion is that it's mostly coming from folks
who aren't distinguishing between something like elegance as a mathematical
work and effective pedagogy.

The first person I met singing its praises was a hardcore linux guy who
insisted on doing every task through a terminal with emacs—and this doesn't
surprise me. I feel like there's a similar aesthetic at play here, and maybe a
bit of fear that doing anything but the toughest option will make them weak
(choosing these things on their own would be insufficient for that
conclusion—but it often comes with a kind of scoffing attitude toward the
'lesser' options).

The logic behind toughest = most effective is a little confusing to me. Sure,
grit has its uses in intellectual work, but getting effective instruction and
building a solid foundation of concepts seems like it would outweigh it.

~~~
llamaz
That's weird - I think Rudin is great and do everything through a terminal
through emacs...

~~~
westoncb
Haha—that's great!

But wait, you weren't in Berkeley, CA in 2013 were you?

~~~
llamaz
I'm from Australia. My theory is that of the subset of people that are
interested in both computer science and math, a significant portion use linux
and if you use linux, then emacs is the best LaTeX editor (auctex and reftex
are amazing). And "doing everything in emacs" is just what naturally happens
when you use emacs long enough.

~~~
westoncb
Makes sense. And so did your explanation for liking Rudin above (or wherever
it's positioned now).

------
ouid
I'm probably in a very slim minority of people who took their first topology
course before they took their first real analysis course, and I think my
understanding of both topics improved as a result of taking them in this
order.

there's no way to actually avoid epsilon delta arguments in real analysis, but
it's helpful to know that there is a more "intuitive" way of thinking about
continuity (although admittedly it's a little weird when you first encounter
it), that requires a lot less algebraic magic.

Anyway, there's more to real analysis than the topology of the real numbers,
but I think it's a great starting place.

------
colejhudson
Hey! Not too long ago I did this myself. Real analysis is somewhat unique in
mathematics for the intuitive writing it's attracted.

The most clearly written is John Baylis's 'What Is Mathematical Analysis?'. I
_strongly_ recommend you read this. It's 125 pages of clarity and intuition
building.

More rigorous alternatives would be: 1\. Elias M. Stein's 'Real Analysis:
Measure Theory, Integration, and Hilbert Spaces'. 2\. Robert S. Strichartz
'The Way of Analysis'

Personally I would use the Baylis book and the other two as reference.

Additional resources include the top voted mathoverflow.com and
math.stackexchange.com answers. Beyond useful.

Lastly, I have a twitter account (@math_twitr) that indexes (mostly) academic
mathematicians. You might want to look through my follows there, they
regularly post useful materials.

Addendum: Hm, in addition, I think you might want to look through this Amazon
wishlist of mine consisting of those math books recommended for clarity by
those who should know:
[https://www.amazon.com/hz/wishlist/ls/2B6H3IG4PS0R1?&sort=de...](https://www.amazon.com/hz/wishlist/ls/2B6H3IG4PS0R1?&sort=default)

------
fofoni
Hi, I'm a maths student in Brazil myself :) I'll answer in english, though,
since I'm not sure about the HN policy on comments in foreign languages.

Can I ask you which university you're from? I'm not aware of many universities
besides UFRJ which offer an "Applied Maths" degree.

About real analysis, I took the summer course in IMPA, an I used only Elon's
books. I really like them (for the books themselves, not just because they're
in portuguese). There are two: the thick one: "Curso de Análise", and the
thin, silver one: "Análise Real" (I like to call them Elão and Elinho :)).
Elão is very detailed and has lots of examples, but mentions topics which may
be too specific and not covered in your course. Elinho is much more terse, and
great if you need a quick summary.

I would also consider reading David Bressod's "A Radical Approach to Real
Analysis". It's an awesome book, which mentions historical motivations for
everything, and has a really different approach to teaching analysis (it will
certainly help you _learn_ analysis, but might not help too much in your
course, since it's quite non-traditional).

If you're not used to proof techniques, I highly recommend Keith Devlin's
"Introduction to Mathematical Thinking".

About strategies to studying analysis: examples. I think it's really important
to work out lots of examples by hand all the time. Every time you read a
definition in your textbook, whatever it is, close the book and try to think
of some examples of mathematical objects which satisfy the definition. When
you're done, try to think of other examples which differ significantly from
the ones you came up with before. When you open your book again, if the author
presents examples, read them with attention. TLDR: as the other comments have
made it very clear, you shouldn't be reading an analysis book without a pencil
on your hand; you should feel active, not passive, while studying analysis.

Finally, I don't know about any youtube channels that could help you with an
analysis course, but you should be aware of Mathematics.StackExchange. It's a
great Q&A website/community; I've asked a lot of questions there while
studying for my undergrad courses.

Wish you the best in your course and you maths career :)

~~~
pedrodelfino
Thanks, fofoni! I am actually doing a double degree in Law and Applied
Mathematics at FGV, Rio de Janeiro. (I know, Law & Applied Math is weird
rsrs). Maybe we should hang out someday. My email is p.delfino01 at gmail dot
com, drop me an email!

~~~
thraway180306
Law+Math degree is definitely on the weirder side of things I've heard (and I
endorsed Oxford's joint CS and Philosophy). But in that case I may allow
myself to make an equally far stretch connecting something lawyerly with real
analysis (that isn't about RA utility in data science for law enforcement).
About how model theory shapes logic. It is advanced (practically algebraic
geometry now) and not really related any more, but the basic issue came from
set theory and analysis: models of infinitesimals, like in Keisler books.
Since then it came to encompass and classify all logic-based mathematics (not
every creative reasoning in mathematics is logical! though papers always are)
and more exotic logics such as the „default logic” sometimes employed by
lawyers.

It's Bressoud BTW, I endorse that too. Along with TW Körner „Companion to
Analysis: A First Second and Second First Course” with Lang or Zorich as base.
I wouldn't be as insistent on pencil at all times if it were to prevent
broader reading or just expanded skimming.

------
jfv
Do NOT read Rudin. He is terse and unless you are already well versed in
mathematics it is simply incomprehensible.

My recommendation would be Spivak’s calculus. There are a million great
exercises and the book is beautifully typeset and overall a pleasure to read.
Don’t let the title fool you, there are analysis exercises in there.

~~~
x0x0
You have to decide between two things:

1 - you want to learn to prove things. Then yes, Rudin is a shitty text to
learn by yourself because he really likes a certain type of, for lack of a
better phrase, "beautiful" proof that requires a bunch of insightful jumps to
get to. He'll then show the proof and really not discuss about how he got
there. What a student needs is the ability to string facts/theorems that he or
she knows together and how to turn that into a proof. Without a good
professor, Rudin is (imo) terrible for that.

2 - you want to learn analysis, and care less about proving things. Reasons
for this may be you need a bit the underpinnings for various reasons, and you
care less about proving things and more about understanding. I think Rudin is
a pretty good text then.

Spivak's calculus is a great book but be prepared to spend a _lot_ of hours on
it.

------
edanm
I highly recommend Francis Su's Real Analysis Youtube lectures, on Youtube
[1]. He is an _amazing_ teacher.

I first started trying to learn Real Analysis from Baby Rudin, but I couldn't
understand the point behind the ideas introduced there. Then I started
watching these lectures, which are based on Baby Rudin and mostly follow it,
and it helped a lot (together with reading the main text itself - a crucial
step).

The only bad thing is that only half of Rudin is covered - the other half is
covered in Real Analysis 2, which is unfortunately not online as far as I can
tell.

[1]
[https://www.youtube.com/watch?v=sqEyWLGvvdw&list=PL0E754696F...](https://www.youtube.com/watch?v=sqEyWLGvvdw&list=PL0E754696F72137EC)

------
throwaway080383
Since you are student at university, I would emphasize, in addition to your
own reading: _Talk to people who already understand the area_. In office hours
with your professor, or grad students, or even more senior undergrads. Just
hang out with people who know more than you and talk to them about math, not
even necessarily just analysis. It may not be the specific things they teach
you, but rather the modes of thinking that are obvious after the fact but
impenetrable before.

------
trendia
In addition to everything else you do, I recommend the following book:

"Counterexamples in Analysis" by Gelbaum and Olmsted.

You will find that many of your intuitions you picked up in calculus are
violated in analysis. For instance, in calculus, many examples are both
continuous and differentiable everywhere. But is every continuous function
also differentiable? Nope! See the Weierstrass function [0].

The book is full of such counterexamples that will help you understand
analysis at a deeper level (and avoid many pitfalls)

[0]
[https://en.wikipedia.org/wiki/Weierstrass_function](https://en.wikipedia.org/wiki/Weierstrass_function)

~~~
rocqua
Really, for me most of my analysis knowledge is based on counter-examples. The
counter-examples motivate the theorems and definitions, help check intuitions,
and sometimes even help me remember the theorems and definitions.

------
totalZero
I wasn't a math student, but I would probably look at the OpenCourseWare from
MIT if I were trying to learn this stuff.

Analysis is 18.100 at MIT -- the variants are called 18.100A, 18.100B,
18.100C. There are further classes in the same vein, as well, such as 18.101.

[https://ocw.mit.edu/courses/mathematics/18-100a-introduction...](https://ocw.mit.edu/courses/mathematics/18-100a-introduction-
to-analysis-fall-2012/)

[https://ocw.mit.edu/courses/mathematics/18-100b-analysis-
i-f...](https://ocw.mit.edu/courses/mathematics/18-100b-analysis-i-fall-2010/)

[https://ocw.mit.edu/courses/mathematics/18-100c-real-
analysi...](https://ocw.mit.edu/courses/mathematics/18-100c-real-analysis-
fall-2012/)

~~~
steindavidb
I taught myself 18.100a from the book and website when I took it because class
was too early in the morning and found them sufficient to make up for the
missed lectures. Wonderful resources. Highly recommmend.

------
yantrams
I'd highly recommend picking up 'Understanding Analysis' by Stephen Abbott for
self study. Beginner friendly and easy to digest yet rigorous. I can't think
of a specific strategy but trying to visually understand the core concepts
like convergence, continuity etc could be of help.

------
matheus2740
I've watched a series of lectures on the subject by Harvey Mudd college, on
youtube [1]. It helped me a lot, tho, I'd have to say these are introductory
level, not really a deep dive, but more useful as an introduction.

[1]
[https://www.youtube.com/watch?v=sqEyWLGvvdw&list=PL04BA7A9EB...](https://www.youtube.com/watch?v=sqEyWLGvvdw&list=PL04BA7A9EB907EDAF)

------
Jugurtha
May I suggest that you prepare mentally to read all the recommended books in
the next ten years or so.

After you get into that mindset, pick one from the curated list and stick with
it.

This will greatly enhance your experience with the book by reducing the
anxiety that you might be reading the "wrong" book and _missing out_ on the
unicorn book out there, which you will read eventually.

In my experience, having that mindset (all vs mutually exclusive) diminishes
the importance we give to the _choice_ of the book, because we'll read them
all. When the _choice_ becomes less important, we spend time actually reading
books instead of deliberating on which books to read.

My book recommendation include:

    
    
      - "A Course of Higher Mathematics" - V.I. Smirnov.
      - "Differential and Integral Calculus" - N. Piskunov.
      - "Problems in Mathematical Analysis" - Demidovich

------
kanishkalinux
Walter Rudin's book on Real Analysis is normally considered as classic in this
area. However, you can also take a look at Mathematical Analysis by Tom M.
Apostol, which helps in developing good intuition about the subject.

------
wish5031
As others have mentioned, Tao has fantastic resources, and his blog might be
worth looking at. Rudin is a classic. I studied real analysis through
“Advanced Calculus” by Fitzpatrick which has some great material.

For more advanced analysis (esp. functional analysis) I would look at Kreyszig
or Hunter & Nachtergaele.

The best way to prepare imo is to just do proofs between now and the start of
the course. Try to find practice proof problems online and see if you can do
them or find an entry-level book on discrete math. Problem-Solving Strategies
by Engel is a good but slightly more advanced book for a beginner.

------
trentmb
Abbotts Understanding Analysis.

------
meiralins
I like the books from Elon Lages Lima ( in portuguese). The small one,
'Analise Real' has suggestions/answers for many of the exercises in the book.

------
gbacon
Remember all the tricks and shortcuts you were taught when learning calculus?

Yeah, forget those.

~~~
pedrodelfino
Interesting. My courses on differential equations (PDE and ODE) were basically
tricks. Very frustrating.

------
starchild_3001
Ha! It took me 3+ years to _really_ understand real analysis. How? I tried to
imagine 10, 20, 30 etc examples for every abtract definition in the books.
E.g. take the definition of open set. Try to imagine 10s of examples of open
sets. Then try 10s of examples of closed sets. And similar. Then in your mind,
you should develop the intuition that "open set is something that looks like
one of these" vs "close set is something that looks like the others" etc. Then
take the definition of continuous function, try to imagine every example
possible! Just work on defn's with many, many, many examples. Henceforth, the
theorems and proofs will become obvious... and presumably you'll end up being
a good theorem-proof style mathematician.

No books necessary! If anything, I liked The Elements of Real Analysis by
Bartle.

~~~
thraway180306
This is sometimes called inquiry-based teaching or Moore method, and… there is
a textbook advertised as adhering to this particular style, Carol Schumacher's
Closer and Closer: Introducing Real Analysis, haven't had it in hand though.

------
phaet0n
This following a fantastic book, with good exercises, and most importantly it
includes applications:

 _Real Analysis and Applications_ by Davidson and Donsig

[http://www.math.uwaterloo.ca/~krdavids/RAA/real.html](http://www.math.uwaterloo.ca/~krdavids/RAA/real.html)

------
stiff
Springer has problem books with full solutions:

[https://www.amazon.com/Problem-Book-Analysis-Books-
Mathemati...](https://www.amazon.com/Problem-Book-Analysis-Books-
Mathematics/dp/1441912959/ref=sr_1_2?ie=UTF8&qid=1521910574&sr=8-2&keywords=analysis+problem+book)
[https://www.amazon.com/Problems-Solutions-Undergraduate-
Anal...](https://www.amazon.com/Problems-Solutions-Undergraduate-Analysis-
Mathematics/dp/0387982353/ref=pd_sim_14_2?_encoding=UTF8&pd_rd_i=0387982353&pd_rd_r=0W0RXBR8HFR8EV0WNY5J&pd_rd_w=BdfFH&pd_rd_wg=fUEbl&psc=1&refRID=0W0RXBR8HFR8EV0WNY5J)

The second one is for Lang's "Undergraduate Analysis" book.

------
sonabinu
Coursera's 'Introduction to Mathematical Thinking' is a great starter course
for real Analysis [https://www.coursera.org/learn/mathematical-
thinking](https://www.coursera.org/learn/mathematical-thinking)

------
chaboud
I've spoken with more than one person who made it through Real Analysis intact
by reading through "Introductory Real Analysis" by Kolmogorov and Fomin.
There's a Dover version that you can probably find for $12 used... It was
where I started, but I know several people who found it invaluable after
struggling with other texts.

Introductory Real Analysis (Dover Books on Mathematics)
[https://www.amazon.com/dp/0486612260/ref=cm_sw_r_cp_apa_a62T...](https://www.amazon.com/dp/0486612260/ref=cm_sw_r_cp_apa_a62TAbNFPVY5M)

Come to think of it, there are a _lot_ of good Dover books on mathematics.

------
corndoge
Like another commenter said, "Understanding Analysis" by Abbot is a fantastic
book. I am not mathematically inclined and this book got me an A- in my
analysis course in undergrad. Rarely is mathematics so clearly explained.

------
johnsonjo
I don’t have any resources for analysis directly to recommend that haven’t
already been said, but there’s some good videos of Calculus by 3blue1brown on
YouTube called The Essence of Calculus [1]. They are really well made and
explain Calculus in a way that you get an intuitive feel for it. It may be
helpful to learning analysis to understand Calculus really well, but I’ve
never taken analysis so I can’t say for sure.

[1]:
[https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53...](https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr)

~~~
praeconium
Yes, 3Blue1Brown is amazing to get the idea behind and have revelation. At
least in my case.

------
jinfiesto
For math majors, baby Rudin is the standard. I would say though that it's
probably too terse for most students. I like Pugh's book quite a lot. I think
it strikes a good balance between not being long winded while providing enough
explanation. Checking the whole answer as you mention is not going to be that
helpful in general. There's more than one way to write most proofs and the
nature of proof problems is that you have the answer at the beginning. Just
make sure you're familiar with basic proof techniques.

------
haZard_OS
Others here have offered some great suggestions already, so I will offer one a
little off the beaten path:

Foundations of Mathematics [https://www.amazon.com/Foundations-Mathematics-
Ian-Stewart/d...](https://www.amazon.com/Foundations-Mathematics-Ian-
Stewart/dp/019870643X/ref=br_lf_m_fa2gmqeu35du8k8_img?_encoding=UTF8&s=books)

This book is meant to help one transition from performing math in an
algorithmic manner to generating proofs based on logic and also set theory.

------
joebergeron
Don't know if anyone has suggested it here (I'd honestly be surprised if
someone hasn't) but Principles of Mathematical Analysis by Walter Rudin,
affectionatey known as "Baby Rudin" is a classic book. It's known for being
relatively difficult and dense if you're just beginning with analysis, but if
you go through the book and complete a fair number of exercises, it's an
incredibly rewarding experience and definitely grants a ton of mathematical
maturity.

------
monaghanboy
Digression: Terry Tao wrote a book on analysis? That's awesome!

Recommendation: [https://www.amazon.com/Introduction-Analysis-Dover-Books-
Mat...](https://www.amazon.com/Introduction-Analysis-Dover-Books-
Mathematics/dp/0486650383/ref=sr_1_2?s=books&ie=UTF8&qid=1521948380&sr=1-2&keywords=real+analysis+dover&dpID=41p-0AKJ7lL&preST=_SY291_BO1,204,203,200_QL40_&dpSrc=srch)

------
auntienomen
There's a Dover book called "Introductory Real Analysis" by Kolmogorov &
Fomin. It's one of Richard Silverman's translations from the Russian. It's got
a few typos in it and the feel is a little old-timey, but the mathematical
content is beautifully laid out. Read it for culture and a look at the bigger
picture. It should be a good complement to Tao's book.

~~~
thraway180306
Also Shilov.

------
throwawaybbqed
Good question ... I haven't seen a great course on real analysis on
Youtube/MOOC platforms. If anyone has recs, I would also be interested.

~~~
ianai
For sure, there’s only so much a lecture in RA can achieve. Ultimately the
student has to write rigorous proofs, and lots of them.

------
VideoEveryDay
Rudin's classic texts are a great resource.

~~~
thraway180306
Rudin is a good breviary rehearsal if one already almost-knows and feels the
material. People with certain inclinations may get a warm fuzzy feeling how
things neatly fit together as if by omniscient design. Otherwise it is a
crossword puzzle to amuse the god himself.

You may get a feeling you understood things (and earned that), but you are
wrong. When I ask people what they really remember Rudin from, what specific
piece of proof they learned specifically from his main and baby books, I hear
only vague answers. There is a pretense of “teaching to think” by omission,
also common in some dated textbooks, but I personally would leave that to
professionals over at philosophy dept. and focus on clear exposition leaving
neatness for examples. OTOH if you are predisposed to lauding yourself for how
smart you (or Rudin) are for figuring all the tricks (knowledge of which stays
at this level), you are in for ego boost (or bust).

Especially for real analysis his treatment of Lebesgue integral is worse than
just about anyone else's I know (also, in 21st century it's time for better
integrals like e.g. Henstock–Kurzweil). The only thing worse is again Rudin's
own treatment of differential forms.

In professional mathematics Rudin is renowed for numerous many things, among
them the Rudin–Keisler order in the theory of ultrafilters and ultraproducts.
It is a sign about reading order. Because as it happens Keisler also wrote a
textbook, and from a diametrically opposed perspective. Equally far fetched in
the other direction, one of intuitionistic non-standard infinitesimals. I
think being a product of certain totalising era of uniformisation in
mathematics these texts are complimentary.

For a reader interested in somewhat extended real analysis I would recommend
Lang, Bressoud, Körner („Companion…”).

Mentioned Terence Tao book from weblog-notes for his original RA course is
also freely downloadable as pdf.

Finally Strichartz is overly chatty wordier antithesis to Rudin.

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markhkim
I'm surprised no one mentioned this yet: make sure to talk to your professor,
frequently. Introductory analysis courses exist primarily to teach a certain
way of thinking, and there is, after all, no better way to learn how to think
than to talk to someone who already knows the ways. Take advantage of what you
have.

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MaysonL
You might check out Polya and Szebot: Problems and Theorems in Analysis: two
volumes. Old (original German edition 1925), but by one of the great
mathematicians of the twentieth century and a longtime collaborator. You might
also like Polya's "How to Solve It", a true classic.

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spacedome
An interesting supplementary book is "Analysis by Its History" that gives
insight into how classical analysis was actually developed. Not so great to
learn from initially, but gives some background on the intuitions from which
the modern definitions are based.

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auvrw
your strategy of working exercises is a good one. it can also apply to results
in the text: attempt to prove the result independently before reading the
proof supplied in the text. if you find this practice enjoyable, it might be a
sign that you'd like to study pure math.

texts: carothers for reading like a novel, rudin for taking apart like a car
engine...

... or, drop all your classes and learn to formulate everything in the terms
of measure theory from the beginning. halmos's texts on any (mathematical)
subject are almost always well balanced ...

... o! that reminds me. also, there's this thing called "functional analysis"
that'll be worth looking into after basic topology is well-cemented.

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makintunde
I think a lot of responses here are forgetting that OP is an undergraduate
student in an _Applied_ Mathematics course. "A Concise Introduction to Pure
Mathematics" by Martin Liebeck would be my suggestion.

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roscoebeezie
One thing that helped me was to draw pictures both before and during proofs.

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uptownfunk
I would recommend baby Rudin, but it was only after a course in real analysis,
measure theory, and functional analysis that I was able to come back to it,
understand it and appreciate it.

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ninguem2
Since you are in Brazil, Elon Lages Lima "Curso de Análise".

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superquest
Draw lots of pictures. That habit helped me learn analysis most.

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ghufran_syed
I hear that libgen.io is a useful resource for books...

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graycat
Part I

One way and another, I got a good background in _real analysis_. So, okay,
I'll try to answer:

An answer depends on what is meant by _real analysis_.

Part of the answer is advanced calculus, and part of that is the Gauss, Green,
and Stokes theorems. If do these the _modern_ and high end ways, then get as
deep as like, and spend as much time as like, in differential geometry,
calculus on manifolds, differential forms, E. Cartan, exterior algebra,
algebraic topology, etc. But for 19th century physics and engineering, there
is a way to get a good treatment of what need in about a nice weekend from
(own TeX markup):

Tom M.\ Apostol, {\it Mathematical Analysis: A Modern Approach to Advanced
Calculus,\/} Addison-Wesley, Reading, Massachusetts, 1957.\ \

Note how old it is! It's no longer really "modern"! Get a used copy; that's
what I did. You don't want a newer edition!

But with that out of the way, might take a fast pass through an old MIT
standard

Francis B.\ Hildebrand, {\it Advanced Calculus for Applications,\/} Prentice-
Hall, Englewood Cliffs, NJ, 1962.\ \

There's a lot of fun stuff in there, but it's TOO short on proofs. So, I'd
pass it up and return later when know enough _real analysis_ to guess or
understand most of the proofs easily.

Note: In (the relatively elegant and easy to read)

George F.\ Simmons, {\it Introduction to Topology and Modern Analysis,\/}
McGraw Hill, New York, 1963.\ \

the flat statement IIRC "The two pillars of analysis are linearity and
continuity."

Well, for the linearity, really need a good background in linear algebra. For
this, you need at least three books, a really easy one, elementary, that
starts with, say, systems of linear equations and Gauss elimination. Then a
more advanced one that emphasizes the axioms for a vector space, does vector
spaces with at least both of the real and complex numbers (over finite fields
can be important for computing but not for _real analysis_ ) and emphasizes
eigenvalues and eigenvectors, and finally the grand one, written at the knee
of von Neumann as a finite dimensional introduction to Hilbert space theory,

Paul R.\ Halmos, {\it Finite-Dimensional Vector Spaces, Second Edition,\/} D.\
Van Nostrand Company, Inc., Princeton, New Jersey, 1958.\ \

If you get very far in _real analysis_ , then you will want a good treatment
of at least basic Hilbert space theory, and Halmos is a good start.

For his section on multi-linear algebra, I'd skip that unless plan to take the
exterior algebra of differential forms seriously. In that Halmos book,
concentrate on vector spaces, vector subspaces, linear transformations,
eigenvalues, eigenvectors, and Hermitian and unitary matrices. Also at the end
note the cute ergodic theorem! The big deal in about the last half of the book
is spectral decomposition -- don't skip that.

For your second book, I used E. Nearing -- he was a student of E. Artin at
Princeton. So, Nearing's book is high quality stuff. I worked carefully
through that and learned a lot. But his appendix on linear programming is a
disaster! Can do nearly everything important in linear programming and its
simplex algorithm as just a simple \-- learn it in an hour -- extension of
Gauss elimination. Nearing does finite cones and dual cones for which he never
makes a clear connection with linear programming. And although he works with
all those cones, still he misses the theorems of the alternative -- Farkas,
etc. -- important in parts of optimization, convexity, etc.

Also recommended is Hoffman and Kunze and, IIRC, available for free on the
Internet as a PDF file.

There is much more in linear algebra, e.g., from R. Bellman, R. Horn, on
numerical methods, etc. but these are not crucial for a rush to _real
analysis_.

Maybe part of _real analysis_ , that is, advanced enough, is "Baby Rudin",
_Principles of Mathematical Analysis_. The later editions have near the end
some tacked on material, mostly without sufficient context, on the exterior
algebra of differential forms. Skip that. If you want that material, then go
for R. Buck, _Advanced Calculus_ or Spivak, _Calculus on Manifolds_ or really
just go for a real book on differential geometry, manifolds, calculus on
manifolds, etc. Such differential geometry is from important to crucial for
several objectives but is NOT on the mainline of a rush to _real analysis_.

So, what is going on in Baby Rudin? Okay, the main idea of the book is that we
can give a solid development of the Riemann integral for a function that is
continuous on a _compact_ set. So, get to learn about continuity, that is, one
of the two pillars of analysis. Then hand in hand with continuity is
compactness, so get to see that. All of that is in just the first few
chapters; that's what those first few chapters are all about -- continuity and
compactness. E.g., get to learn that in R^n (for the set of real numbers R and
a positive integer n), a set is compact if an only it is closed and bounded --
super, important, crucial stuff, the key to a clean up of Riemann integration,
that is, material Newton didn't know.

Then with that material on continuity and compactness, Rudin does the
Riemann–Stieltjes integral. There, mostly just ignore the Stieltjes part with
its possibility of _step_ functions (maybe as a cheap answer to what the
physics people try to do with the Dirac delta function, which, of course, is
not really a function, but has a clean fix-up with distributions and measure
theory) and just read that Rudin material for the Riemann integral of first
calculus. You will likely never see the Stieltjes extension again.

The main idea is: A function continuous on a compact set is also, presto,
bingo, wonder of wonders, really nice day, _uniformly continuous_ , and that
makes the derivation of the Riemann integral really easy. Really, that's the
core idea of the whole book. Baby Rudin can seem severe, but with this
introduction you should be able actually to like it a lot. Later in the book,
Rudin touches on the fact that the uniform limit of a sequence of continuous
functions is continuous -- same song, next verse. It was a Ph.D. qualifying
exam question for me; I did get it.

Later he gives a really nice treatment of Fourier series -- that is very much
worth reading.

I'd suggest one side trip: Cover the inverse and implicit function theorems.
They are just a local, non-linear generalization of what you will see really
easily for linear transformations in linear algebra via, right, just Gauss
elimination. For a source? There is a good treatment in W. Fleming, _Functions
of Several Variables_. IIRC, there is also a cute proof based on contractive
mappings.

So, by then you will have a good start on both continuity and linearity.

~~~
graycat
Part II

Starting there, as you go on in _real analysis_ , you will place much less
emphasis on continuity. The main work will replacing the Riemann integral by
something that does much better on the _edge_ or pathological cases.

Well, the Riemann integral was, right, a lot about _area_. Well, it's actually
not a really good theory of area -- doesn't have all the nice properties we
would wish for. So, that view of _area_ gets polished up and replaced by
_measure_ theory. Next, that theory of area is used to define the Lebesgue
integral, which is much nicer than the Riemann integral, e.g., gets rid of the
assumptions of a compact set and continuity.

Lebesgue was a student of E. Borel and did his work near 1900. You will see
the Heine-Borel theorem in Baby Rudin.

In simplest terms, the Lebesgue idea is to partition on the Y axis instead of
the X axis like the Riemann integral does. Just why that works so well is
cute. It also frees up the domain of the function to be much more general, and
in 1933 A. Kolmogorov used that fact, finally, to give a good foundation for
probability theory, the one accepted now in essentially all advanced work in
probability, stochastic processes, and mathematical statistics.

Likely the nicest book to read on _real analysis_ is H. Royden, _Real
Analysis_. But also, later, read the first, real half of W. Rudin, _Real and
Complex Analysis_.

Note: For _real analysis_ , complex valued functions of a real variable are
important; functions of a complex variable are quite different and not
important; Rudin has some deep reasons to disagree with me. Rudin is no doubt
correct; for students trying to learn, I'm correct!

Once you have the main ideas in mind, and maybe what I've given here will be
enough, Rudin is more succinct and nicer to read than Royden. But read them
both, Royden first. Go through Royden quickly since will do it all again in
Rudin.

In Royden, notice his Littlewood's Three Principles as a cute view of what is
going on in measure theory and the Lebesgue integral. Royden also has some
exercises on upper and lower semi-continuity -- glance at those and get the
main idea for the remaining connection with continuity. Notice the nice
treatment of differentiation -- the connection between integration and
differentiation is weaker for the Lebesgue integral because the integral is so
much more general. Then notice the Carathéodory extension result \-- that is
the key to defining measures on the real line, in particular, Lebesgue
measure.

Spend at least a weekend in

John C.\ Oxtoby, {\it Measure and Category:\ \ A Survey of the Analogies
between Topological and Measure Spaces,\/} ISBN 3-540-05349-2, Springer-
Verlag, Berlin, 1971.\ \

An amazing weekend.

Keep at hand

Bernard R.\ Gelbaum and John M.\ H.\ Olmsted, {\it Counterexamples in
Analysis,\/} Holden-Day, San Francisco, 1964.\ \

Notice that for the Lebesgue integral, there is one, central proof technique
in just four steps: Prove the theorem for a function that is just a positive
constant defined on a set, say, just a box. By linearity, generalize to finite
sums of such functions. Then by monotone continuity (see early on the monotone
convergence theorem), prove the result for all non-negative functions. Last by
linearity again, get the result for all _integrable_ functions.

Okay, for _integrable_ , there is a cute approach, trick: Define the integral
for non-negative functions, even ones defined on the whole real line or all of
R^n (for the set of real number R and a positive integer n) or an abstract
measure space. Okay, easily enough, that integral can have value positive
infinity (the Riemann integral gets sick here). Then do the same for a
function that is all <= 0. Then given a function, write it as the sum of its
positive part and its negative part. Integrate those two parts separately.
Now, if at least one of those two integrals is finite, then add them for the
integral of the given function. So, we are okay in all cases except where the
positive part integrates to positive infinity and the negative part integrates
to negative infinity -- we don't want to subtract infinities because
permitting that ruins the laws we want for arithmetic.

For this four step proof technique, can knock off a really nice version of
Fubini's theorem, that is, interchange of order of integration.

The Lebesgue integral with the dominated convergence theorem, which will see
early on, give nice versions of differentiation under the integral sign.

Note: Commonly in physics, engineering, probability theory, etc. we write
integrals over the whole real line. Alas, really, for Riemann integration, the
appropriate theorems are rarely presented; they are not in Baby Rudin. Really,
the Lebesgue integral is needed. So, you've been needing the Lebesgue integral
for a long time.

Then get to apply the Lebesgue integral to the powerful Radon-Nikodym theorem
(Rudin gives von Neumann's cute proof based, amazingly, in part on just
polynomials), get the main, important duality theorems (the Lebesgue integral
is the main linear operator!), Banach spaces, Hilbert spaces, and the Fourier
integral.

Cover that and can claim you have a good start on _real analysis_. You will
also have apparently so far the only good background for probability,
stochastic processes, and mathematical statistics. E.g., the Radon-Nikodym
theorem is crucial for conditional expectation, Markov processes, martingales
(amazing things with one of the strongest inequalities in math and a super
short proof of the strong law of large numbers), ergodic theory, sufficiency
in statistics, a quite general proof of the Neyman-Pearson result in
statistical hypothesis testing, and more, e.g., the role of Brownian motion in
potential theory, stochastic optimal control, ....

Go for it!

~~~
Bahamut
Royden is a classic - my mom and I both used it in our grad math programs,
almost 30 years apart or so.

------
ianai
I worked through 9 chapters of Apostol.

