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.
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2$\begingroup$ Is this question not more appropriate for Math.StackExchange? $\endgroup$– JP McCarthyCommented Oct 9, 2014 at 12:08
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$\begingroup$ relevant other question. matheducators.stackexchange.com/questions/1302/…. I love Kenneth Ross's book as an introduction. It is meant to bridge the gap between elementary calculus and baby rudin. $\endgroup$– WetlabStudentCommented Nov 27, 2014 at 2:41
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4 Answers
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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.
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$\begingroup$ 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 $\endgroup$ Commented Oct 8, 2014 at 14:35
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$\begingroup$ I've edited the answer to include a book. $\endgroup$– Chris CCommented Oct 8, 2014 at 16:08
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$\begingroup$ 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. $\endgroup$ Commented Oct 8, 2014 at 17:55
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$\begingroup$ @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 ???? $\endgroup$ Commented Oct 8, 2014 at 18:38
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1$\begingroup$ 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. $\endgroup$ Commented Oct 8, 2014 at 21:33
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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.
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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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1$\begingroup$ To expand your answer, would you mind adding why you recommend Pugh's book? $\endgroup$– J WCommented Oct 9, 2014 at 17:18
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1$\begingroup$ 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. $\endgroup$ Commented Oct 9, 2014 at 18:34
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$\begingroup$ math.stackexchange.com/questions/50444/…, the first anwser of this thread includes some good reasons for choosing Pugh's book. $\endgroup$– user2139Commented Oct 10, 2014 at 1:22
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$\begingroup$ @user2898908: Then I suggest including the link and a short summary in your answer. $\endgroup$– J WCommented Oct 10, 2014 at 5:18
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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.