What should i study next after this?

Question

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

Answer 1

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.

Answer 2

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.