Need some advice on analysis books

In summary, the individual has recently completed an elementary analysis course using Spivak's Calculus and received an A due to the professor's generosity with grading. They are currently studying Folland's Advanced Calculus text and Rudin's Real and Complex Analysis, but are finding the latter to be difficult. They plan to continue studying Rudin's text but have also been recommended to use Ahlfors' Complex Analysis text for additional practice. They have also been advised to start with Rudin's Principles of Mathematical Analysis before attempting the more advanced text. The individual has some prior knowledge in complex variables, but it is limited to basic concepts and theorems.
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I've just finished an elementary analysis course using Spivak's Calculus. I did get an A in the course but only because the professor was often very generous with the grading. The course I am in currently is using Folland's Advanced Calculus text.

At the same time I have also been studying from Rudin's Real and Complex Analysis which has been moving along really slowly. I find it exceedingly difficult but I still do plan to forge on ahead with it. I asked my professor about another book for more practice and he pointed me to Lar Ahlfors Complex Analysis text. Can anyone comment on Ahlfor's text and compare Rudin and Ahlfors? I am doing this all on my own. I will occasionally ask questions here on physics forums but obviously if the text is too hard it won't help me any as I am already struggling a lot with Rudin.
 
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i haven't used spivak myself, but how much analysis was covered in your course?

for instance, did you learn about compactness, cauchy sequences, completeness, convergence, continuity in metric spaces and normed spaces? it seems to me that going from spivak directly to rudin's Real and Complex Analysis is a pretty big jump. if you aren't able to do the exercises, then maybe you could have a look at rudin's Principles of Mathematical Analysis which may be easier for you to go through since you already have some analysis under your belt.

for ahlfors, have you taken a complex variables class already? ahlfors is also pretty terse so it may be difficult to go through without prior knowledge on the subject.
 
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For analysis, I did learn about compactness, cauchy sequences, completeness, convergence, and continuity in metric spaces. The stuff like compactness, completeness, and metric spaces didn't come from Spivak. I learned by basically teaching myself those things. The reason I am attempting to work through big Rudin is because it started as a reading course with the professor. We didn't get very far but I thought I'd continue it anyway. Now that you mention it, I will definitely put down big Rudin and spend a month or so going through baby Rudin first.

As for complex, I only know the most basic things like for example the theorems that have to do with sequences and series in the real plane that can be used in the complex plane. I was also exposed to the proof of the fundamental theorem of algebra and stirling's formula.
 

Related to Need some advice on analysis books

1. What are the best analysis books for beginners?

Some great analysis books for beginners include "Calculus: Early Transcendentals" by James Stewart, "Introduction to Real Analysis" by Robert G. Bartle and Donald R. Sherbert, and "Understanding Analysis" by Stephen Abbott.

2. Are there any analysis books specifically for self-study?

Yes, there are several analysis books that are great for self-study, such as "Real Mathematical Analysis" by Charles Pugh, "A Course in Real Analysis" by N.L. Carothers, and "Principles of Mathematical Analysis" by Walter Rudin.

3. What are some good analysis books for advanced learners?

For advanced learners, some recommended analysis books include "Real and Complex Analysis" by Walter Rudin, "Functional Analysis, Sobolev Spaces and Partial Differential Equations" by Haim Brezis, and "Measure Theory and Integration" by Michael E. Taylor.

4. Can you recommend any analysis books that focus on applications?

Yes, some analysis books that focus on applications include "Real Analysis: A Constructive Approach" by Mark Bridger, "Mathematical Analysis: An Introduction to Functions of Several Variables" by Mariano Giaquinta and Giuseppe Modica, and "Complex Analysis for Mathematics and Engineering" by John H. Mathews and Russell W. Howell.

5. Are there any free online resources for learning analysis?

Yes, there are many free online resources for learning analysis, such as OpenCourseWare courses from universities like MIT and Yale, YouTube channels like MathDoctorBob and MathTheBeautiful, and websites like Khan Academy and Brilliant.

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