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i'm hoping that some of you might be able to help. i'm currently in a state of perplexity and doubt.

i'm finishing up my first year at graduate school, in math. the problem is what to read over the summer. by now i have discovered that i'm most interested in analysis and applied analysis. (my second interest would probably be geometry.) differential equations and dynamical systems interest me.

i would like to learn basic techniques that are used by practicing analysts and applied mathematicians. i once read a book on smooth euclidean manifolds (munkres'), and i have never ever used the material on manifold integration and differential forms that i thoroughly studied. i don't like this.

as far as my background, i read and solved most of the problems in munkres' texts in topology and analysis on manifolds, and i've been able to read most of the first half of rudin without skipping a beat, likewise with dudley's real analysis and probability. i would like a book with good problems, but appropriate, like those in munkres' books.

so far i'm looking at coddington and levinson's book on linear ODE's but i'm worried that since linear DE theory is already completely understood (i don't remember where i read this), i'll end up never using what i learn here. can anyone comment on this?

also, i'm looking at rosenlicht's undergrad analysis book. i know i'm "beyond" it, but the material is so well-presented and the exercises are very good (i've heard some are very difficult), that i want to go back and re-read it and solve most of the problems. should i not bother, since i'm already about to go into my second year of graduate school?

can anyone here recommend any good books in analysis, applied analysis, and dynamical systems (introduction)? (i decided i won't read rudin nor dudley's book anymore, because there is too much conflict with my personal taste.)

i'm also looking at kreyszig's functional analysis book; it seems interesting but many people seem to think it's not at all challenging.