# Order of study in mathematical analysis textbooks.

I have just started studying mathematical analysis and, when choosing the books I was going to use, I realized that there is some levels of difficulty.

I am very lost in the sense that I don't know which is the proper order of study.

The idea I have right now is the next:

• 1: First course in Real Analysis (epsilon-delta approach): Here is where we can find those books on 1 real variable such as calculus (spivak), understanding analysis (Abbott), Analysis I (Terence Tao), A course in mathematical analysis I (Garling), Mathematical Analysis vol. I (V. Zorich), Introduction to Real analysis (Bartle)...

• 2: Second course (using concepts such as metric spaces..., multivariable calculus, vector calculus): Here we find books like The elements of Real Analysis (Bartle), Analysis in Euclidean Space (Hoffman), Rudin's PMA, Pugh's Real Mathematical Analysis, Apostol's Mathematical Analysis, Analysis II (Terence Tao), Mathematical Analysis vol. I (V. Zorich), other books in several variables (such as Lang's, Fleming's...), Real Analysis (carothers), some introductions to Lebesgue integration...

• 3: Intermediate-Advanced: Here we have those books with a hard content in Topology (metric spaces, topological spaces, Continuity, Compactness, Completeness, Connectedness, Convergence...) and Analysis in several variables using all these topological concepts(introduction to differential forms, vector analysis, analysis on manifolds...). Some books in this section are: A course in mathematical analysis II (Garling), Topology (munkres) and other books in Point-Set Topology (as well as books dealing exclusively with metric spaces or topological spaces), Foundations of Modern Analysis (J. Dieudonne), Introductory real analysis (Kolmogorov & Fomin), Mathematical Analysis vol.II (V. Zorich), Introduction to Topology and Modern Analysis (Simmons), calculus on manifolds (spivak), analysis on manifolds (munkres)...

This is the classification that I have in mind but it is correct ? I am just in the first level but, due to I am self studying the subject, I would like to know how to study all these topics in the most ordered possible way.

Finally, I have 2 additional questions.

• Which are the main differences between the 2 books from Bartle ? (Introduction to real analysis and the elements of real analysis) When should I study each ?
• Is vector calculus the same than multivariable calculus (as seen in the second course) ? Is there 2 kinds of multivariable calculus (one studied in the second course and called 'vector calculus' and another in the intermediate course and called 'vector analysis' ? )

• Analysis in metric spaces is just a generalization of analysis on the real line $\mathbb{R}$. In fact, the real numbers are a metric space. Nov 22, 2017 at 17:36
• Euclidean $n$-space can be viewed as a metric space, but there are examples of metric spaces can stand alone as abstract spaces on which one can define continuity and the like. Baby Rudin has a very nice treatment of metric spaces. Nov 22, 2017 at 17:47