Beneficial to touch theoretically deeper texts earlier in core areas e.g. analysis, algebra?

Question

Desired future direction: Dynamical System(Chaos), PDE

More beneficial to read theoretically deep, modern and masterpiece texts earlier, (e.g. levels like UTX/GTM/GSM/LNM/CSAM) ? Especially in core areas (e.g. Analysis, Algebra, Topology etc.)

i.e. Read briefly less for intro-level, then quickly drop to the deeper/modern texts, although basic skill's not trained enough, but to be trained again/better in 'deeper-stage'. (Sometimes it seems learning new material in deeper/complex 'environment/context' gives better/faster maturity.)

e.g: Calculus/Analysis: Apostol -> Hardy, Courant, Stein, Rudin, Amann

Linear Algebra: Axler -> Dym, Grueb, Lax, Halmos

Algebra: van der Waerden -> Lang, Hungerford, Jacobson, Bourbaki

Topology: Kelley -> Milnor

Answer 1

"for topic of calculus, I spent long time on many books of intro level" definitely sounds like a mistake. I would not advise anyone to read many introductory calculus textbooks. (Unless you serve on an undergraduate committee or plan to write a textbook of your own.)

It's hard to find anything wrong with reading deep and modern masterpieces early. Some amount of struggle and confusion is healthy. There is a simple way to find whether such advanced reading is beneficial: try to explain what you read to someone else (e.g., to an imaginary blog reader :).

Answer 2

The best books are self-contained, which is one reason they are so widely used. It's nice that you can assess what you know and don't know.

Rather than give a philosophical answer, I will make a few suggestions of things that I have used as a self-studier that I highly value. Especially for the insight and clarity of their presentation. They have all been used at great math teaching institutions.

For real analysis this free download is lecture notes by Fields Medal winner Vaughan Jones when he was at Berkeley

https://sites.google.com/site/math104sp2011/lecture-notes

I also like Pugh's "Real Math. Analysis," which he also used for an honors section.

For linear algebra - Axler's "Linear Algebra Done Right"

For algebra you can watch these videos of Benedict Gross at Harvard

http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra

The accompanying text is Artin's "Algebra" which I prefer for the first exposure to groups and rings. If you want to go on, I would move to Dummitt and Foote.

These books are all designed to introduce you to the material and build to take you quite far.

I would not give a thought to to any label (like GTM) the point is to really learn the material which is it's own reward.

Answer 3

You should read what is appropriate for your current skill level. If I had to break it down into percentages, I would recommend spending about 70% of the time reading sources that are right at your skill level, 10% of the time reading something at an easier level than your skill level for the purpose of refreshing yourself and seeking complete mastery, and 20% of the time reading more advanced materials with the intention of trying to get a sense of what is to come and to push the limits of what you are presently capable of.

But the main thing is to keep working hard to learn more. That formula for success is pretty robust relative to the specific path you take as you acquire more knowledge and skills.