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I am in the middle of a slightly ambitious attempt to learn Analysis on my own. I skimmed through Rudin(Baby), Chapman Pugh, William Wade, Stephen Abbott and Strichartz and ended up preferring the Elements of Real Analysis by Bartle over the others. This might probably surprise you but I am someone who prefers the classics and I enjoy the systematic presentation and the enjoyable writing. More importantly I like the fact that the whole book is based on the space $\Bbb R^p$ and does not extend to general metric spaces which I think will help me in a first course (so that I can picture the arguments geometrically).

I only just finished the Topology chapter. But (maybe I'm wrong) looking at some questions posted on this site I can't help but wonder if there are enough exercises in Bartle's book or if the ones in it are sophisticated enough. I can't really go for another book for the problems since everything I've found is either based on metric spaces or just the real line.

So I would love it for someone who is familiar with Bartle to tell me if there is another supplementary text with more/ better exercises or if I actually need one?

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    $\begingroup$ Wade's book spends very little time in a general metric space setting. There is only one chapter on it and you can skip it and still do the rest of the book. You could look there for exercises. $\endgroup$ – Joe Johnson 126 Feb 28 '14 at 15:50
  • $\begingroup$ @Joe Johnson 126: Thanks for that. I just checked it out at the library. You're right. The first half of the book is mostly based on the real line and the second half on $\Bbb R^n$. Plenty of exercises too. Guess my initial skimming was lousy. Thanks again.. $\endgroup$ – Ishfaaq Mar 1 '14 at 2:18
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    $\begingroup$ Spivak's gem Calculus on Manifolds leaps to mind. $\endgroup$ – Andrew D. Hwang Mar 11 '14 at 19:50
  • $\begingroup$ I really don't understand how the introduction of the word "metric space" and the "$d$" instead of norm can prevent you from thinking like if it were $\mathbb{R}^n$ for intuition. $\endgroup$ – Aloizio Macedo Mar 12 '14 at 3:03
  • $\begingroup$ @Aloizio Macedo: Any advice is appreciated. Do you think I should just not worry about it and go for any decent Analysis book meant for a first course? $\endgroup$ – Ishfaaq Mar 12 '14 at 9:36
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I suggest two real classics. The first one is Introduction to Calculus and Analysis Volume I and II from Richard Courant. The first volume is based on $\mathbb{R}$, the second treats Analysis on $\mathbb{R^d}$. Metric spaces are not considered.

If you are looking for lots of good examples from Analysis without (explicitely) using the concept of metric spaces then you could read Problems and Theorems in Analysis Volume I and II from Pólya and Szegö.

They are also available as ebooks in the Springer publishing company.

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  • $\begingroup$ Any ideas on my perturbations? As in if I actually need a supplementary book for exercises? I know doing more problems can't hurt. But am I worrying over nothing in presuming Bartle's exercises aren't sophisticated enough? Would like to ping @Chris Leary about this too.. $\endgroup$ – Ishfaaq Mar 12 '14 at 0:17
  • $\begingroup$ Thanks for the bounty and dont't worry. I only wanted to provide you with a comprehensive source of interesting examples. It's informative to browse through them and you could select a few to study them intensively. The main thing is to extend your experience by working through them and you have a lot of fun! :-) $\endgroup$ – Markus Scheuer Mar 13 '14 at 14:35
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You might consider looking at "Analysis in Euclidean Space" by Kenneth Hoffman. The book concentrates attention on $\mathbb{R}^n$, and has some good exercises. Generalizations refer to normed linear spaces, generally $\mathbb{R}^n$ with some norm, or $\mathbb{C}$, the complex numbers. This is great preparation if you eventually decide to get into functional analysis, where complete, normed, linear spaces are a central topic.

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