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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)...

  • Advanced (measure theory, advanced analysis): Royden, Folland, Papa Rudin, others...

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' ? )

Thank you in advance.

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Your "Levels" 1 and 2 look right to me. I've heard that Tao's first analysis book is good, but I have never used it personally. I started with a book similar to the ones you listed in level 1 and then moved on to Rudin's Principles of Mathematical Analysis (Baby Rudin). Baby Rudin also covers functions in several variables.

For the upper level material, I highly recommend Royden and Fitzpatrick's Real Analysis, Fourth Edition. Not only does it cover measure and integration theory, it also has sections dedicated to metric and topological spaces. Daddy Rudin (a.k.a. Rudin's Real and Complex Analysis) is pretty advanced; I would recommend that only after you've worked through a book like Royden's.

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  • $\begingroup$ But I am more interested in the theory than in the books itself (by the way, thank you for your recommendations). I mean, I would like to know the different levels that there is from a first course to an advanced level. For instance, in first place we make mathematical analysis with the epsilon-delta approach, then I don't know very well where to go. What does it mean to make real analysis over n-Euclidean Spaces or Cartesian Spaces or Metric Spaces... ? $\endgroup$ – Diego Garcia Nov 22 '17 at 17:30
  • $\begingroup$ Well, you basically captured it pretty well. Start with the basics of continuity, differentiation, and integration, then look at those concepts via metric spaces (maybe some topological spaces), and finally move on to the more general theory of measure and integration. $\endgroup$ – Sir_Math_Cat Nov 22 '17 at 17:35
  • $\begingroup$ 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. $\endgroup$ – Sir_Math_Cat Nov 22 '17 at 17:36
  • $\begingroup$ And what does it mean ' over euclidean spaces ' ? Is the same than 'cartesian spaces' ? What is the difference between these two and 'over metric spaces' ? Are they more o less general ? I ask you this because in the second course I mentioned two books which work over euclidean and cartesian spaces (Hoffman's Analysis in Euclidean Space and Bartle's Elements of real analysis respectively). And why can we find books called 'Real Analysis' but that work over metric spaces ? $\endgroup$ – Diego Garcia Nov 22 '17 at 17:43
  • $\begingroup$ 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. $\endgroup$ – Sir_Math_Cat Nov 22 '17 at 17:47

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