Terence Tao's Analysis I and Analysis II Book Review
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- Опубліковано 2 сер 2024
- In this video I go over two amazing books written by Terence Tao.The books are Analysis I and Analysis II.
These are the books on amazon
Analysis I: amzn.to/2T310oF
Analysis II: amzn.to/2SNpnI5
The links above are my affiliate links. As an Amazon Associate I earn from qualifying purchases, so please use my links if you want to support my channel:)
For more book reviews check out my playlist
• Book Reviews
I have a PhD in Mathematics. After 50 years of studying math It is still a mystery to me why most textbooks do not provide solutions to the exercises. Even the relatively few great math books would be vastly improved if they were “complete”.
Definitely
Just getting ready to ask the same question. If working as many problems is a key to learning math, well, knowing that the correct answers to all problems exist might encourage those learning to work more math. This should be a criteria for all math textbooks.
it is Fermat who created and initiated this tradition
@@taopaille-paille4992 What an insight!
@@standowner6979 Haha, And in all seriousness, I think exercises without solutions are helpful. At the start of my studies in maths I was always looking at the solutions because I felt too stupid and not able to find solutions, but then I started to really work and try to solve them, and it is what made me progress a lot. In the Boubarki books, you ll find a lot of exercise without solutions, so instead of Fermat maybe this tradition has to do with the Bourbaki
I studied those books and they’re brillant and worth every day I spent my time with them. Big recommendations! 📚
👍
Thanks
When you are studying Algebra Two does it matter how old the book is and for what state standards the book was written to satisfy? I am refering to the algebra that we learn in America in highschool in the k-12 education system?
Been binging all your videos, you have a great channel my friend. Keep up the amazing content. You got me back into math and I just purchased ‘calculus made easy’ because of your recommendation.
Dude you are a really book lover as me, i v collected a lot of books for mathematics, algebra, topology,so on. I'm study mathematical analysis, good luck to you
That's awesome!
Never Being
Can you contact with me I want to ask something about Mathematics.
That discussion on the Russell paradox makes me recall the Simmons' book on topology.
Definitely got me excited to use these to refresh (and really learn this time!) analysis. Side note: Companies like fed ex will rebind books for less than 10$ if you want to replace the cheaper paperbacks with a spiral binding
Nice review and good exposition about this great book by Prof Tao, a field madelist mathematician.
Love from India.
Thanks a lot for the videos.
Thanks sir for making these amazing math book videos!
I am so grateful that you created a mathematics channel; you say a lot of stuff that math teacher don't say. I have been very benefitted from your channel.Please keep sharing math videos
Thank you
"I read and understood it" says a lot about how smart you are also...
Well the guy does have a PhD in mathematics
The average iq of such a person is 150. That sets you in the top 0.1% for intelligence.
@@robertmorrison1657 Thats just not true
@@MonkeyDLuffy-gd6se What is the average iq for someone with a PhD in Mathematics?
@@robertmorrison1657 Ive got no statistics on that but very unlikely to be 150.
@@MonkeyDLuffy-gd6se If my statistics are not accurate, it would be keen for the both of us to find some research on the matter.
Seems like a good book for intro analysis or honors calculus. I read the one by Elon Lages Lima (relatively unknown since it's only available in Portuguese and Spanish) on my first year of undergrad and it covers more or less the same topics including limsup and liminf, but it lacks the motivational prelude this book has -- that's very nice on Tao's behalf, especially since he has a really clear view of the "big picture" of maths.
portugues?
@@vpambs1pt eu falo espanhol (sou chileno), mas entendo um poquinho
@@technoguyx Engracado ^^
Grande el Lima viejo
@@vpambs1pt Sim, existe vida inteligente que fala português.
My copy just arrived and I really like it a lot. I have only two complaints. The first is that (in my opinion) he leaves too many of the theorems as exercises. And the second is that he doesnt cover Stone-Weierstrass and Arzela-Ascoli, theorems i guess i thought were core results in advanced calc that should be covered. But otherwise, the book is excellent and does a great job providing intuition behind seemingly complex results. I really liked his section on multivariable calc, and I thought he did a really good job explaining implicit function theorem, which really confused me before this. Anyways, thanks for the recommendation.
Awesome!
do u have any other recommendation for books?
@@forsaken7161Baby Rudin chapters 1-7 and then Munkres Analysis on Manifolds for the multivariable stuff . Those are what I used for my advanced calc courses, but Rudin in particular is extremely dense and difficult to understand if you are new to the subject. So maybe Abotts Understanding Analysis would also be good to have for that material, but his treatment of metric spaces is kind of brief in my opinion
@@tsshamoo2376 thank u for your quick reply. I will have a look into it
A great vid as always
This is THE greatest book on undergraduate analysis I've seen
Yes it reads really well doesn't it!!
Understanding analysis by abbott
@@froglet827 Tried it and Tao is million times better
Taxi cab or L1 is used extensively in statistics in the form of the LASSO model. You can also draw a parellel between the median in taxi cab and the mean in euclidian.
The way this book is setup is literally the exact same as the books that my university give me, which they type up themselves. like the overall look of this book is literally the exact same. Guess it's due to it being written in latex
awsome channel.........it makes me remember my past days ...................keep it up
Thank you!
Thanks for the review math sorcerer! I decided to study from this book, but was wondering whether the 2nd part covers introductory Topology material as well? or do I need a separate Topology book for that.
Great book review. I'm sorely tempted after watching your video. Do you/does anyone know if any of the answers to the exercises are published or at least discussed online? That's the only concern I have about getting this book, lack of an answer key. Cheers!
Is that the famous Terence Toa field medalist? Nice book review.
Would you ever go through each book chapter by chapter, through the reading and the exercises. That would be so awesome, and worth more views than all the justin bieber and nicki manaj videos combined.
I get so excited watching your videos. Your description of mathematics, especially analysis, is so elegant . I wish i could read an analysis book like that.
Did you learn analysis from lectures in college, or by self-teaching by reading?
I wonder what is the best method. Like the famous mathematical quip, there is no royal road (no shortcut) to geometry, every person has a unique academic journey.
Thank you! I took lots of classes in college. I did self study too but college yes😄
And yes that is him and good idea thank you!!
Great Review!
thank you!
They are such a good books that I literally had to print the pdf I bought and I'm keeping it on the table(the only books there). They introduced me to such a different world with all the formalising right from the basics. I haven't even conpleted it, because it's just so packed with interesting thoughts. One thing to note is that these books only deal with real analysis(I hope I'm right on this). It's a timeless book, but somehow this not the standard of how the subject is taught and the book isn't even that popular, should get waay more press imo.
❤️
If you have read his book on competition math you would have never doubted his writing skills how approachable he makes the subject rare person with genius and writing ability
Tao is a very good communicator. Modest, fluent, responsive, considered, honest, and humorous. Very good person, a great scholar and a gentleman to the core.....
Definitely. I have seen some of this videos here on UA-cam, where people interview him. He seems like a really good person👍👍👍
I really think he is, he is one of the good guys and a brilliant mathematician 👍
I have been reading them and now im interested in his book on measure theory,
Oh cool I should check that out
There are many better books on Measure Theory than Tao’s one. Bartle’s book and Cohn’s book are way better.
@@davidgendron6955 You are a mathematician???????????????????
A math student, yes.
@@davidgendron6955 So, you are not enough level to remark
Wow tao has an undergrad analysis (aka inequality theory) text??? I have his measure text INCREDIBLE
Nice video. I'll soon be buying a copy of Terry Tao's two Analysis books. Although Terry Tao is of Chinese descent, he's actually from Australia, which may partially explain his excellent exposition style in writing. Another reason is his high intelligence and mastery of the subject matter. This brings to mind another master that wrote very readable Calculus books a couple generations ago: Richard Courant. His two volume Calculus set is a real gem. His second edition of the set was co-authored with Fritz John, but these volumes were not as much a pleasure to read.
I still don't have the Courant books. I looked at them years ago but don't own a copy. So the first edition is the way to go?
@@TheMathSorcerer Times have changed and the amount of material taught in Calculus courses has greatly increased. Furthermore, publishing technology now allows easy inclusion of graphs and other instructive illustrations into mathematics texts. I have found that these illustrations are helpful in clarifying concepts and in avoiding misconceptions. The two volume Courant calculus texts were delightful reads in their day. [I am referring to the first English edition, not the far more massive second edition with Fritz John. There may have been an earlier version in German, as was the case with Hilbert and Courant's Mathematical Physics two volume set.] Another delightful read was Higher Mathematics for Students of Chemistry and Physics by Mellor. I read this book in my mid-teens and could hardly put it down. Some of the notation has since changed and a more contemporary version of Mellor's approach was published in the UK about 30 years ago, but it lacked the magic of Mellor. By contemporary standards, I'd have to recommend standards such as Thomas' text or Bruce Edwards' text because of the many thousands of illustrations. These tomes are about 1200 pages each. I also can recommend the three volume calculus set by Jerry Marsden. There is also a three volume set on solutions to the problems, making it a six volume set. [Jerry Marsden passed away in 2010, and CalTech, where he taught, made his text available for free online. Jerry was a great mathematician, a genuinely nice person, and very generous in sending out reprints during the snail mail era.] Gilbert Strang's three Volume calculus book is also worthwhile. At a more challenging level, Paul Bamberg and Shlomo Sternberg's two volume set, Mathematics for Students of Physics and Engineering, is worth considering, but not intended to be a relaxing read, but rather a challenging read, but a worthwhile tome for your library. There are many other worthwhile texts, and this is but a few that come to mind. There are others that are also worthwhile.
Thank you thank you!!! Good suggestion s will look into these😄
Finally, thanks for the review, they're awesome!
Could you give me your opinion? I have a background on proofs Linear algebra and analysis 1 and 2
I'd like to self study Abstract Algebra what book do you recommend for self study and good with proofs? Not to dry/harsh but one that can make a better mathematician? I'll have Abs algebra next semester, so I'd like to study it before
Saracino's book is the easiest I think. It's not perfect but it is easier than many others😄
@@TheMathSorcerer Thanks
Probably a bit late, but I love Aluffi's book.
Followed, perhaps, by "Analysis on Manifolds" by Munkres?
Good video, btw. I see the Indian publishers coming up with better and better content.
The only things I look for in books is readability.
If I understand something, I can try to proove it. This books builds understanding after which most proofs become easy.
Are you a book hoarder? Well, I am considered as such by my parents. Damn, I wish I have another room for books. I collect books on mostly digital art, graphic novels, programming, etc. I have few on maths. Now, I use Kindle and Open Library.
Have you tried libgen.is? It is a huge library (obviously not legal but ...) where you can download a lot of math books!
Yes book hoarder for sure! I have sooo many bookshelves.
@@TheMathSorcerer same
Great sharing! Ive just order this awesome collection from amazon .mathlover from HK😊
Nice!
@@TheMathSorcerer just receive my math package from amazon:) cant wait to read it
Maybe I will buy those
I gave both books a try and really liked them, in particular the examples after each definition. But the lack of solutions for the exercises is kind of a letdown. Especially when the proofs for some propositions and theorems are swapped out as exercises. Sometimes, some authors are demanding to much from the (even advanced) readers.
download the Slader app and I thought you can find almost all the solutions there.
@@iankovski what's the full name of the app? i only see apps for drawing
Going to have to put this on my wish list
I'm determined I can find this for a good price, I'll have to keep my eyes out! 👀
👍
At least in India, the Hindustan Book agency version is much chepaer than the Springer counterpart (they're the exact same book, published jointly in a sense).
I took a course on the analysis I do you think that these two books would be good to continue to self teach and learn integration theory?
What book similar to Analysis II which has exercises AND solutions do you recommend? Cheers!
Can I use this book for the first few chapters as someone who is just now learning calculus?? I’ve only learned precalc
These are the books I referred to when I decided to study analysis more deeply (I'm a control system engineer ). I enjoyed them a lot. They make me reflect on this question: when I can say that my proof or solution is right?
I learn about taxi cab on this book more than in taxi companies.
Haha
I have an electrical engineering background but have always had a fascination for mathematics. I had obviously taken calculus, linear algebra, differential equations, optimization theory courses, etc., but from an applied maths point of view. Do you think I will be able to grasp the full essence of these books ?
Yeah definitely , if you have even some slight proof writing skills you can get a lot from this based on your background
Should I read this, or abbots understanding analysis before reading taos books
I took Calc classes based on Apostol books and took a Proof Writing class based on discrete math. Is this enough foundation to take ANALYSIS with Tao's books? And for baby Rudin ?
What would you say about Mathematical Analysis by T.M Apostol?If one completes both volumes of T.M Apostol's Calculus,then what would be better choice among Rudin's and Apostol's Mathematical Analysis?
I like apostols mathematical analysis
Great suggestion. Cheers.
It's a great book
Will check it out thank you 😄
The table of contents looks exactly the same as most german analysis books. The Königsberger, the bible of german analysis, is structured exactly the same.
Oh wow very cool!!
hey math sorcerer , could u do a video on explaining the order for someone to study the Terence Tao books? its the only logical consistent book I have found so far , which matches my level of understanding , thankiew !!!
actually I find the rudin analysis I extremely unreadable , some of his argument doesn't logically follows , hence the chain of reasoning is broken into pieces and I found this problem among most the accessible books for me
Sure👍
looking forward to it , THANKSSS
Can one use these books without taking calculus before? Will these textbooks be able to give the knowledge usually given in the calculus course?
I think that would be difficult, you could but you would not understand a lot of it. I say try it and just expect not to understand, it's ok not too and you will probably understand some of the stuff😀😀
I trust your opinion (and thank you for the book review). I'll be buying a copy soon.
“I cannot recommend this book any more”:
Isn't it funny how even a simple sentence in ordinary language can have two directly opposite meanings!? 😉
lol
You must get e review on zuckerman number theory by wiley publishers
Ok!
Hah! Had this text in my Analysis I and I walked away very bruised (and withdrew from the class that semester)! English is my second language and this book just smeared me!
Intro to Analysis by William Wade went infinitely better for me.
Nice hehe,I should check out the Wade book👍
Why does these things have to be so difficult for me still.. I just want a good math book to read and understand as a very early beginner, something that isn't advanced calculus
Ya it's hard for everyone, I have a playlist on advanced calculus check it out 👍
do u have good book on non euclidean geometry, topology and related topics ?????????????
What level of calculus is needed to understand this? Like is it calc 1, or 2 or that. Thanks in advance
Some proof writing experience is required , that would be better👍
@@TheMathSorcerer thx a lot :)
📙💯
Terence Tao is the best living mathematician in the world.
His Brain at 3000 Hz, Mouth at 60 Hz, A Genious.......
@@gilloon9964 Who ? Never heard of him
How much it costs
it is a basic requirement for anybody to pursue a phd in stats or applied math i believe.
Why would analysis be useful in applied mathatic
@@robertmorrison1657 it is not a question of usefulness.
This publisher is really good, the books are cheaper and SEWN. So these are actually of higher quality than the original Springer books.
Tq sir
These books look extremely interesting.
Alas I am now just too old for books on mathematics.
Yes Tao is absolute genius 🙂
the set theory problems in volume 1 are too hard..i couldn't get them after trying for ages. there is no proof that -(-x)=x in chapter 4 on integers/rationals, nor is it given as an excercise. Wondering why this was? It would have made things easier imo. Otherwise, i really liked the book, managed to do most of the problems and it took about 500 pages of scrap work to get through them all haha. The well defined and trichotomy proofs were pretty tiresome. I like how the proofs of important theorems are often left partly as an excercise, proving the existence of least upper bound was one of my favourite problems in the book, and it was split up into an entire section of problems at the end of the chapter.
👍
I really want to self study this book during quarantine but I can't find solutions to the book anywhere, how can I know what I am doing is right?
Yeah I have not seen a solutions book unfortunately. Maybe I will try or rather, I really really should make a playlist and work out some solutions from this book!
Thanks, I really enjoy Tao's writing style, it just sad that I have very few exposure to proofs yet so it kind of unsettling not knowing the right answer.
Mathematical Analysis was one of the first courses I did as an undergraduate... It was strange for most of us because it set that boundary line from high school math ( which is mainly problem solving) to Undergraduate Math (which is mostly about abstract proofs )
wow what an intense jump from highschool, so hardcore!
@@TheMathSorcerer its the standard in Europe
What other books would you recommend to study Analysis?
Fitzpatrick and also buck
@@TheMathSorcerer piskunov?
Im a little late here, but for me Pugh's Real Mathematical Analysis is still the best. It goes REALLY far at times(man the topology chapter is wonderful for a beginner math student, its SO good) and it has very good problems and the exposition is top notch.
Why did i find this video just after my real analysis exams😭😭
Good book but I still prefer Rudin if you are somewhat familiar with the intuition behind analysis for the sheer reason that there are virtually no solutions available for Tao’s texts. This is problematic as Tao leaves a sizable amount of material for the reader to prove. He is, however, great at building the intuition.
I have them in e-book version. They are not good for deep studying. Maybe an introduction of what exists out there for analysis. Also, all the exercises are the proofs of the basic theorems (very bad tactic in my opinion).
Tao is gifted.
for sure, super gifted
This looks amazing, now I kind of want it. lol.
solid book:D
@@TheMathSorcerer sounds even better by the minute.
Nice
thank you! it's an awesome book:)
I had always heard Manhattan metric instead of 🚕 metrc.
I wonder why the author would want two different types of covers for the same series.
10 weeks wait time for me to get the book in Belgium :(
wow!!!
just check allian connes book of trim ...................and please tell what it is ...............im from theoretical physics not from maths
Hey, I'm starting Analysis 1 at my university, we're using Calculus by Spivak. Do you think this book is more introductory than Spivak's?
This book is more advanced than spivak. Spivak is awesome though, it's cool you actually get to use it for a class, very lucky!!
The Math Sorcerer I see, I'm asking because the material and content-wise, Spivak's book seems very straightforward (maybe it's because I have taken Discrete Math course before), but the problems seem very difficult. I am starting in fall, but the course is infamous for being the hardest first year course. I have to take this to switch to pure mathematics program. People who take this course say that its is far more difficult than discrete math. Is this an exaggeration or should I be worried?
@@ericryu599 eyo, how did the course go?
@@aayusharya6899 I'm still taking the course. It's a year long course. We're on integrals right now. I'd say by far, Chapter 7 on "Three Hard Theorems" is the hardest chapter. Spivak actually proves the three threorems in Chapter 8 just because it uses the notion of infimum and supremum. In my opinion, it's sort of a mix between dry read and casual read. But to be fair, there are no analysis books that are 'fun' per se (at least none that I've encountered).
Spivak is a deceivingly difficult book if you tackle all the problems. It’s an excellent book though!
How good are the examples? When a definition or a theorem statement is complex or not obvious, good examples are vital. A good example not only states the example, but explains why it's an example, demonstrating how the example fits the definition, for instance. Not everybody does this.
Last week, I picked up a book on matroid theory and I couldn't quite grasp the book's first definition of a matroid. I already knew that a matroid was some kind of generalization of a vector space. The only example of the matroid definition the book cited was of course a vector space. The trouble was that there was little else besides the statement that vector spaces were examples of matroids. That was absolutely useless to me, or anyone else who needed to wrap their heads around the definition. So, how does Tao describe his examples? Does he merely state them? Or does he show how the examples meet the requirements of the definition?
Why is it that there are commonly no answers in the back of analysis books?
I know that happens a lot
@@TheMathSorcerer Does anyone know if there is a specific reason to do that? Or are authors just lazy? (just kidding) I find it difficult to study a book without any feedback on whether my solutions work
I think they do it on purpose because they want students to figure the questions out. Also lots of teachers assign those problems as homework so they don't provide answers. This is just my theory.
I thought we have other publishers publishing these two books outside India instead of HBA
I'm really starting to HATE it! (oops) when there aren't answers available for SELF STUDY (which is the best way in my opinion), when you have several books with lots of problems, but not nearly enough answers! Why not? Because I use the answers to challenge myself, just peaking if I'm stuck, getting a little more understanding then trying on my own from that point, then perhaps completing the rest of the problem correctly on my own. So what gives? Just give us some answers. It's unbelievable!
You should read a book called Math: a discrete introduction, by Dr. Scheinerman if you haven't before. I don't think I've ever come across such lucid writing before in mathematical writing.
Oh interesting, thank you !!
@@TheMathSorcerer Yeah, he used to work at the math department in my school. You may come to like it. He has a unique sense of humor.
Awesome
Some time ago a book came out titled, "The Tao of Physics". I suspect Terence is single-handedly the reason why there will not be a book called the Tao of Mathematics.
Is this book fit for a recent high school graduate? I’m familiar with calculus but not multivariable calculus. I’m interested but I don’t know if I should give it a go!
You can, but I recommend multivariable first.
The book is good
yup:)
Which part(1 or 2) is slightly better
@@1stlullaby484 I don't have the part 2 yet
@@TheMathSorcerer which part did you find better (even if it's in slightest)
Free with springer membership
Just because someone has won the Fields Medal does not mean that they are the best textbook writers .Analysis I and II make a confusing subject even more confusing to the average student ..The best American intro Analysis book is "Introduction to Real Analysis" by Bartle and Sherbert .Ross isn't that bad either ....This subject can be presented from a million different points of view .Why should Terrence Tao's point of view be taken as Gospel?? I have taught an intro to Analysis class at a mid sized American University , and even on a good day ,I could not expect the Whole class to understand why x (squared ) is continuous using an epsilon delta argument ...........
believe the hype damn
Being good at math is really a god- gifted thing
talent comes into it but also a lot of work
The books are terrible quality, but damn is it readable. Spend a day reading Tao and spend a day reading Rudin and I guarantee you +95% of students new to the material will prefer Tao. It's so so so digestible it's honestly uncanny.
😮
And ive just decided to write the 3rd volume called....Go learn the 1st two books and then get back to me.....no, I'm not Irish lol
Please keep the camera a bit far.
how about a calculus book with no solutions. fail.
This is good book, but still for me best analysis book in baby Rudin,very rigour text.
Ya that is a rigorous text!
Well i tell my opinion, I hope objective.
Analysis I: Real number and Fields very verbose and confused (4 is not 0, for example)
No Metric Spaces and nor topology, so forget about ε and δ
30 and more pages of introduction useless
To me much more complicated than Rudin
You have to take notes from this mess and it's not easy
And nothing new, only old stuff in a complicated way
Even volume II, quite standard, no innovations no gifts.
There are no examples, no good exercises, no gift at all.
A good book of theory, and not the best
What a pity (California...)
but... I must learn English first :(
Did humans invent or discover mathmatics?
That is the question!