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I am a girl currently in 12, and i have read upto the Fundamental Theorem of Integral calculus in riemann integration, and also solved lot of problems. I have thourough knowledge of $\epsilon - \delta$ proofs and i have studied everything in Real Analysis(whatever i have done till now) in the greatest detail i could. Should i start studying metric spaces now or lear those Second Mean value theorems and Measures, etc. Before that? Should i start studying terence tao II analysis now?

Thanks.

The reason that i am not sure whether to do analysis II is that Terence does not use Cauchy Criterion for ANY of the problems, he either does it by majorization of functions or defining a lot(which make it unncessarily complicated, i have done a different book too in anoher language, but the proofs are relativeky short and easier due to usage of Cauchy criterion, also it mentions the relation between the Darboux and Riemann intergals(which Terence doesn't), and the most confusing thing is that Terence complicated the proof of the fact that product of two riemann integrable functions are also riemann integrable a LOT, whereas it is easy using the Cauchy Criterion as mentioned in the other book, while elementary metric spaces are ipnot introduced in that one while Terence includes it. Thats why i am confused, do i need to do the proofs in Terence's way ONLY?? They are unnecesarily overcomplicated

Sruti Sarkar
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  • What is your goal? – Bram28 Apr 27 '20 at 15:20
  • And what kind of stuff interests you? – Dasherman Apr 27 '20 at 15:24
  • I love to self study and i am preparing for chennai mathematical institute and ISI Bangalore too – Sruti Sarkar Apr 27 '20 at 15:25
  • I love abstract concepts in mathematics, especially rigoruous proofs, difficult theorems, and extremely theoritical concepts, and thus i like real analysis so much. – Sruti Sarkar Apr 27 '20 at 15:26
  • If you're already comfortable with Tao's Analysis I, then certainly Analysis II would be a reasonable next step. But you have tons of good options. You could even branch out of math and read Introduction to Electrodynamics by Griffiths or The Elements of Computing Systems by Nisan and Schocken or Molecular Biology of the Cell. The world is your oyster. – littleO Apr 27 '20 at 15:28
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    I want to stick to math – Sruti Sarkar Apr 27 '20 at 15:29
  • I have detailed my question pleas e check it – Sruti Sarkar Apr 27 '20 at 15:36
  • Have you also learned techniques of integration? To compare, in Stewart’s Calculus text, the Fundamental Theorem of Calculus is presented in Chapter 5, but he continues covering integration for the next two chapters. If you can, have you read most or all of a specific text? At least to give us a baseline of what you have read (and understood). – Clayton Apr 27 '20 at 15:39
  • I think this might interest you. –  Apr 27 '20 at 15:40
  • Instead of committing just to Tao's books, I recommend reading multiple real analysis books simultaneously and focusing on whichever ones you're enjoying the most or finding to be the most clear. It's good that you are following the maxim "don't just read it, fight it!" and trying to find the simplest and most clear proofs rather than just accepting the way the author does it. (On the other hand, Tao usually does things in a very elegant way, so if his proofs seem overly complicated I would wonder if in fact there is some good reason that Tao has chosen to do it this way.) – littleO Apr 27 '20 at 15:40
  • And you should see Prof Pugh's book on analysis as well. Personally I find it easier to read then books by Prof Tao/Rudin –  Apr 27 '20 at 15:42
  • Whats the books name? – Sruti Sarkar Apr 27 '20 at 15:43
  • Real Mathematical Analysis by Charles C. Pugh –  Apr 27 '20 at 15:48
  • Okay sir, i will try Spivak and Pugh's books, are they okay for my level (who has done only upto riemann integrals till now?) Also i see topology in Pugh's book :( – Sruti Sarkar Apr 27 '20 at 15:53
  • If you don’t want to do analysis you can start abstract algebra for a change. I strongly recommend Dummit & Foote. – Invincible Apr 27 '20 at 15:55
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    I want to master real analysis first, i still have time to do abstract algebra later and i will surely try out Dummit & Foote! – Sruti Sarkar Apr 27 '20 at 15:56
  • Yes I think Pugh's book is suitable for you. You can also simultaneously read books on Linear Algebra, for example Linear Algebra with applications by Prof Gilbert Strang. –  Apr 28 '20 at 04:34
  • Yeah, i have read upto the rank nulliy theorem from Axler's book, i dont find Strang's interesting as it focuses more on matrices than theory. – Sruti Sarkar Apr 28 '20 at 06:14
  • Does this answer your question? Books for high school students starting on college math –  May 15 '23 at 23:58
  • Duplicates https://math.stackexchange.com/q/127814. – user1147844 May 30 '23 at 07:22

2 Answers2

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For a person just passing out of XIIth class (India), and interested in higher Mathematics of the pure kind the following books should be suitable:

  1. Real Mathematical Analysis by Charles C. Pugh
  2. Topics in Algebra by I. N. Herstein
  3. Combinatorics and Graph Theory by Harris
  4. Proofs and Refutations: The Logic of Mathematical Discovery by philosopher Imre Lakatos

All these books are written in a somewhat casual style, while maintaining the appropriate standard of rigor. So the probability that a XII class student will lose interest by the writing style is low. Regarding (4), I feel that it contains principles which are usually never expounded clearly in the standard textbooks but experienced mathematicians/logicians just seem to know them.

Another book, with an absolutely opposite style (and when I mean absolute, I mean absolutely absolute) is Prof Landau's Foundations of Analysis. It contains results detailing constructions of number systems which are usually not expounded in detail in analysis books. Although Prof Tao's book Analysis I has done it, so you may be aware of the same.

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Now would be a good time to deepen your knowledge of the real numbers. First study how the real numbers are constructed from the rational numbers (i) Cauchy and null sequences of rational numbers. (ii) real number defined as equivalence class of Cauchy sequences of rational numbers modulo null sequences (iii) proof that a Cauchy sequence of real numbers converges to a real number (iv) definitions of bounded, open, closed, connected , compact sets of real numbers, definition of supremum, infimum.(v) proofs that a set of real numbers bounded above has a supremum, a set of real numbers bounded below has an infimum, a set of real numbers is compact iff it is closed and bounded, the only non-empty connected sets of real numbers are $\mathbb R$, the intervals and the singletons, the continuous image of a connected set is connected, the continuous image of a compact set is compact (vi) applications of the above to definitions, existence and continuity of powers, roots, exponential (defined by its power series) and log functions (ln defined as inverse of exp). Then deepen your knowledge of series-first learn more about finite series (i) 'factorial powers'$x^{(n)}=x(x-1)...(x-n+1)$ and how to convert between them and ordinary powers (ii)forward difference notation $\Delta f(n)=f(n+1)-f(n)$(iii) summation of finite series by anti-differencing (iv) summation by parts. Then study infinite series -as well as the standard tests for convergence/divergence learn Raabe's test, which is very useful when the ratio test fails.Then study power series. As well as the the topics of radius of convergence, adding, multiplying, differentiating and integration, you should look at the algebra of formal power series, especially reciprocation.

P. Lawrence
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  • I haven't done radius of convergence till now. Honestly i find series to be boring compared to continuity, differentiability, sequences and integerability – Sruti Sarkar Apr 27 '20 at 16:31