Book Recommendation for CSIR-UGC NET exam I've to clear this exam (NET Mathematical Sciences) in May, 2019 and I can't take coaching classes for the preparation of this exam as I can't afford it. Can you all pls recommend me best books for this level of exam?
Here's the syllabus and the sample question paper.
I've been studying Axler's Linear Algebra Done Right for some time but it's mostly theoretical, so it'll take a long time to finish this book. I don't know if it's the best book for this exam.
Or should I follow some other book? What about for Real Analysis? Is Stephen Abbott's Understanding Analysis enough?
Please recommend all the books for this exam.
Thank you very much!
 A: Welcome to MSE!
First I explain What is CSIR-UGC NET ?  It is a competitive exam  for Awarding Junior Research Fellowship for doing Ph.D and Eligibility for Lecturership in India (The reason for mentioning this is to clarifying other users) 

1)  Structure of NET: 

First of all, note that the syllabus of this test includes the following topics as you know:

Unit I: Real analysis, Linear Algebra
Unit II: Complex analysis, Abstract algebra, Topology
Unit III: Ode,  Numerical analysis, Calculus of variations, Linear integral equations, Mechanics
Unit IV: Statistics

To Clear this  Net Mathematical Science, you must familiar with atleast FOUR topics and personally I recommend the first two units.

2) Books recommended:

For Linear algebra, Axler book is fine but I also recommend Friedberg's Linear Algebra. Solve as my problems as you can from this books. I also mention two familiar problem books in linear algebra

  
*
  
*Paul Halmos: Linear algebra problem book
  
*Fuzhen Zhang: Linear algebra: Challenging problems for students  

For real analysis, first work through Abbott's analysis or A basic course in real analysis by Kumereasan  and then move on to Rudin. Baby Rudin book contains lot of  challenging problems comparing to Abbott. For Problem Books, I recommend,

  
*
  
*AMS Problems in Mathematical Analysis Book Series(3 books)
  
*Asuman G.Aksoy, M.A. Khamsi: A problem book in real analysis

For Abstract algebra, I think Gallian is enough for CSIR NET. Anyhow, work first Gallian and then move on to Herstein or Artin. Solve as many problems as you can in these books. Also Gallian web page contains lot of resources for algebra.

3) Others:

At each time CSIR  asks repeated type of questions like irreducibility, uniform continuity, use of identity theorem,diagonalizability, etc. (not many and not too little)  
So solving previous years questions is necessary not sufficient! 
From my point of view, trying to answer from Part A is just a waste of time. For preparing this exam, I advice you that do not try to buy a previous years solved books from online. Think and solve your own. Of course, this take lot of time, but that is not the problem. For solving exercise problems, remember, hints and answers in the book back should be consulted only after serious attempts have been made to solve the problems. If you ignore this
advice(not mine, actually Kenneth Ross from Elementary analysis), you will only cheat yourself!. Do not try to learn the different topics in isolation.
Last, Don't forgot the following  two quotes for learning concept and solving problems:

From Axler:  
You Cannot expect to read a mathematics the way you read a novel. If you zip through a page less than an hour, you are probably going too fast 
From Polya:  
There is a grain of discovery in every solution of the problem.

Good Luck!
