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What topics should I read before studying mathematical analysis?

I want to have a solid foundation in terminology, notation and concepts in general.

Please suggest titles for books.

J W
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Eng_Boody
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    Is this question not more appropriate for Math.StackExchange? – JP McCarthy Oct 09 '14 at 12:08
  • relevant other question. http://matheducators.stackexchange.com/questions/1302/lesson-plan-to-self-teach-real-analysis-to-student-with-comp-sci-background/1414#1414. I love Kenneth Ross's book as an introduction. It is meant to bridge the gap between elementary calculus and baby rudin. – WetlabStudent Nov 27 '14 at 02:41

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This is my first answer here, so I hope I'm not breaking any inside rule in this answer.

Your question is too vague in my opinion: Mathematical Analysis is a very wide area of mathematics and is extremely hard to answer it without knowing a bit about your background.

If you already have taken a calculus course you probably can start studying real analysis and Rudin is a timeless classic, although I like very much Real Analysis: An Introduction, by AJ White, with its well crafted exercises that will leave a lot of work to do on your own.

There is no formal pre-requisite, but it will help a lot if you have some experience with one-variable calculus, series and sequences of real numbers.

Jonas Gomes
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To begin your analysis journey, perhaps a book like Abbott to introduce you to the basic topics and some basic proofs for typical analysis classes.

For a rigorous analysis course (think Baby Rudin), I believe that a good grasp of the concept of proof and set theory is required. The little bit of familiarity with topology (particularly that of the topology of $\mathbb{R}$) that you need will be covered concurrently in many analysis books. I've not used the book, but I hear How to Prove it is a good introduction to the transition from solving problems to proving theorems.

To move up to real analysis (Big Rudin, Royden, etc), the basics covered in Baby Rudin should be sufficient.

For topology, I find that Munkres is a great read and has many wonderful problems. Again, confidence with proof tactics is required.

These are essentially the books I used in classes progressing through analysis.

Chris C
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  • what book of topology should i read in and what are the prerequisites of that book assuming i had only taken calculus and linear algebra so far – Eng_Boody Oct 08 '14 at 14:35
  • I've edited the answer to include a book. – Chris C Oct 08 '14 at 16:08
  • From the preface of Principles of Mathematical Analysis: "This book is intended to serve as a text for the course in analysis that is usually taken by advanced undergraduate or by first-year students who study mathematics." I assume first-year means first-year graduate and advanced undergraduate, at the very least, third year undergraduate. I don't believe the OP is at the required level to read it. If I am wrong, I wish him to be more specific. – Mark Fantini Oct 08 '14 at 17:55
  • @ChrisC you mean 1)proves 2)Topology 3)mathematical analysis ?? that's all no other prerequisites and after those 3 steps i can go to real analysis ???? – Eng_Boody Oct 08 '14 at 18:38
  • @End_Boody Perhaps you should be more specific. Do you want to learn measure theory and integration, real analysis in one variable or many variables or something else? Real analysis is a huge area. – Mark Fantini Oct 08 '14 at 18:52
  • @Mark Fantini My final goal is signal processing and that requires complex analysis which requires first real analysis that's what i got from searching the internet, learning real analysis needs some basic concepts found in book titles "mathematical analysis" .... all what i have said here is a result of searching the internet ..if am saying something wrong feel free to correct it – Eng_Boody Oct 08 '14 at 19:32
  • @Eng_Boody That's a lot of important information. Add it to your original post. Your real question is "what is the mathematics needed to delve in signal processing". You don't need to go through so many courses to get the ideas. I'd risk saying that what you need from complex analysis is not a full blown analysis course but comfort manipulating complex numbers, integrals, computation of integrals using residues and conformal mappings. – Mark Fantini Oct 08 '14 at 19:34
  • @Mark Fantini I Tried reading in applied complex variables (the same subjects as u said) but i always encounter the percise definition of limits and neighborhoods .. which needs some digging downwards ... even in standard calculus books i encounter the precise definition, nothing enogh is written in calculus books about that definitions , it implies going through real analysis as i found from searching the internet – Eng_Boody Oct 08 '14 at 19:41
  • @Mark Fantini As the question stands, I fail to see the issue. Yes I agree that it depends on the end goal, but as stated that goal is to study analysis. I believe that the new question covers your concern, but whether they should be merged is another question altogether. – Chris C Oct 08 '14 at 21:11
  • @End_Boody Usually I see topology and into. analysis as done concurrently, but any topology you need to get through the into. analysis will be covered by the analysis book as necessary. – Chris C Oct 08 '14 at 21:15
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    I've edited the question so it's salvageable and your answer still applies. I don't think this question should be merged with "mathematics necessary for signal processing". Also, I don't like Rudin for a first exposure to real analysis. The quote strengthens my view. – Mark Fantini Oct 08 '14 at 21:33
  • That is fair. I now remember back to when I first saw it and it was a simple class. I added a book for a first look at the subject. – Chris C Oct 09 '14 at 00:13
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In theory, there is very little pre-requisite to study introductory level real analysis (sometimes it's called advanced calculus, although this term is also seen applied to multivariable/vector calculus). Things as fundamental as number system (specifically real number) will be rigorously defined.

Practically, I think having the following will help a lot.
1. Some exposure to math proof in other courses (e.g. linear algebra, discrete math, etc.)
2. Non-rigorous version of calculus

For textbook, I also recommend Pugh's Real Mathematical Analysis. http://www.amazon.com/Mathematical-Analysis-Undergraduate-Texts-Mathematics/dp/144192941X

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    To expand your answer, would you mind adding why you recommend Pugh's book? – J W Oct 09 '14 at 17:18
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    As much as I like Pugh's book (engaging writing, well designed diagrams, excellent and very interesting problems), my guess is that it's probably too advanced for the OP. I recommend looking at Victor Bryant's Yet Another Introduction to Analysis, which I've written about several times elsewhere. – Dave L Renfro Oct 09 '14 at 18:34
  • http://math.stackexchange.com/questions/50444/teaching-introductory-real-analysis, the first anwser of this thread includes some good reasons for choosing Pugh's book. –  Oct 10 '14 at 01:22
  • @user2898908: Then I suggest including the link and a short summary in your answer. – J W Oct 10 '14 at 05:18
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Since you ask what to read before studying mathematical analysis (emphasis mine), might I suggest Lara Alcock's recent book: How to Think About Analysis (2014, OUP). She has written a very accessible and helpful guide to introductory analysis, replete with study advice and how to overcome common points of difficulty. It would go well prior to or alongside a textbook such as Abbott or Bryant.

Edit: I was also interested to read about "Self-Explanation Training" on pp. 39-43 as a way to aid understanding proofs. See http://setmath.lboro.ac.uk and http://homepages.lboro.ac.uk/~mamji/files/JRME_SelfExpl_Paper.pdf for further details.

J W
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