# Transition from introduction to analysis to more advanced analysis

I am currently studying intro to analysis and learning somethings about basic topology in metric space and almost finished the course . I am thinking of taking some more advanced analysis. Would it be demanding to take some course like functional analysis or real analysis only with knowledge of intro analysis course ?Do the courses need more mathmatical knowledge to handle?

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It might help if you include the text you are currently using. – amWhy Nov 29 '12 at 3:28
@amWhy Apostol Mathematical analysis on chapter : Basic topology, limit and Continuity And rudin Mathematical analysis as well. – Tom Nov 29 '12 at 3:31
The best person to ask would probably be the instructor of the course you are thinking of taking. Second-best might be the instructor of the course you are currently taking. – Robert Israel Nov 29 '12 at 3:37

Most of analysis requires very little algebra. If you have covered a course in basic analysis, say using Rudin's Principles of Mathematical Analysis, then you are ready to take a course in more advanced analysis, say using Rudin's Real and Complex Analysis, which covers both measure theory and linear operators as well as holomorphic functions.

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But i didn't use the whole book in the course. We just focus on the chapter about basic topology , limit and continuity and sequences in the book of Rudin. Would it be enough to study real analysis? – Tom Nov 29 '12 at 3:37
In your opinion, do you think funtional analysis or real analysis more demanding or more difficult? – Tom Nov 29 '12 at 3:44

I'm currently taking a graduate functional analysis course having only taken introductory analysis (I majored in physics). It's manageable, but knowing measure theory and lebesgue integration would have definitely helped.

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What do you learn in the introductory analysis course – Tom Nov 29 '12 at 3:35

Depending on what you mean by "intro to analysis," a real analysis course covering, say, Lebesgue integration would probably be reasonable, but a functional analysis course would often assume you've been through a basic graduate analysis course already. This all depends on where you're studying.

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I have mentioned above, i have learnt some basic topology to metric space, so i know some basic topology and also the fixed point theorem and so on – Tom Nov 29 '12 at 3:30

Once you have a basic intro to real analysis, you can go several ways.

One would be to cover analysis in $\mathbb{R}^N$ and heading towards analysis on manifolds using books like Rudin's "Principles of Mathematical Analysis" (chapters 9 and 10), or Munkres' "Analysis on Manifolds".

Another direction you can take, if you have covered the Riemann integral in basic analysis, is to study measure and integration theory. You could use the first of Fremlin's free books on measures and integration. I don't have a very good idea of introductory books for this.

If you also know some linear algebra, you could also go on into functional analysis. There are some good books meant for students with a modest background in real analysis and linear algebra. One very popular book is "Introductory Functional Analysis" by Kreyszig. This book neither assumes nor uses Lebesgue integration. If you got as far as sequences of functions in basic analysis, you could also use "Beginning Functional Analysis" by Karen Saxe (this one first gives a self-contained intro to Lebesgue integration).

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