When I come across a notion from algebra or number theory which I don't know I usually check Keith Conrad's page to see if he has written something about it. Key features of his articles are a very clear exposition and carefully worked-out, well-chosen examples. Furthermore, he explains why the definitions are the way the are (why do we require a subring of a unital ring to have the same multiplicative identy as the original ring?).

I am now looking for articles of similar style and quality explaining the basic notions of analysis (both real and complex analysis) and probability theory (so they should be aimed at undergraduate students). More specifically, I mean the notions (and central theorems) you would expect to learn in any undergraduate course on the subject at a German university. The content descriptions of the courses in the following document might give you an idea of what that means: mathematics.uni-bonn.de/study/master/files/MA_QualTest.pdf The courses are: Analysis I & II (page 2), Analysis III (page 3), Introduction to complex analysis (p. 4), Introduction to Probability & Stochastik Processes (both p. 6).

I tried to find something using google and the only thing I found is http://www.mtts.org.in/expository-articles. There are articles on theorems from analysis on that site, but they focus on proofs and do not contain enough motivation for definitions or interesting examples (as far as I checked, I did not read them all). The articles should not be too dense (to give you an idea of what that means: They should not be as dense and short as the articles in the Princeton Companion).

Note that I am looking for articles which are available freely online.

I am aware of the related question Elementary Papers at ArXiv, however, I ask for articles explaining ideas and notions from analysis and probability theory only and do not require the articles to be posted on the arxiv.

  • $\begingroup$ I like this question for two reasons: 1) I find such articles a great bedtime or airplane/train/bus reading, and 2) because I like to share them with interested undergraduates, and I think they really appreciate such articles. Unfortunately I don't have an answer for you, but I am hoping someone will. I'm interested also. $\endgroup$ – user2093 Jan 13 '12 at 17:34
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    $\begingroup$ Should this be CW? $\endgroup$ – kahen Jan 13 '12 at 18:05
  • $\begingroup$ I hope this is *not* made CW. This is a nicely phrased question, and making it CW will rob the OP [and the answerers] of some well-deserved points. $\endgroup$ – Srivatsan Jan 16 '12 at 14:59
  • $\begingroup$ Do you only want online resources? The Princeton Companion is actually pretty decent for this, in my opinion. $\endgroup$ – Willie Wong Jan 16 '12 at 16:06
  • $\begingroup$ Also, what do you consider to be "basic notions" in analysis/probability theory? $\endgroup$ – Willie Wong Jan 16 '12 at 16:08

I think that these notes by John Erdmann might satisfy your requirements for at least a portion of the analysis material. First, there is:

A Problem Text in Advanced Calculus

This set of notes covers basis aspects of topology, basic aspects of integral/differential calculus and concludes with the inverse and implicit function theorems.

There is also:

A Companion to Real Analysis

These notes could perhaps be considered as a continuation of his problem text. More topology is discussed, Lebesgue integration, Banach/Hilbert spaces and concludes with some operator theory. There is also some probability thrown in for good measure, pun intended.

For multivariable analysis/calculus, I also like Jerry Shurman's notes that you can find here. He covers some of the same ground that Erdman does, but goes much farther into "vector calculus" including differential forms and Stokes/Gauss' theorems. Like Erdman's, these notes are also extremely well-written.

Also, speaking of professors named Conrad, you might want to look at Brian Conrad's notes on differential geometry found here. Although these notes cover more DG than you may be interested in, he gives a full accounting of Stokes' theorem and the algebra/analysis that is required to get there. The treatment is considerably deeper than that found in Shurman's notes. Each topic is contained in a separate document and there are probably about 50 or so of these documents, usually between 10 and 20 pages apiece.

  • $\begingroup$ The problem with the books by Erdmann and with Shurman's book is that they are books. I am not looking for a book or lecture notes but articles explaining one idea carefully with many examples and motivation provided. Brian Conrads notes are good, but as you mentioned they are on differential geometry mostly (I am still thankful that you mentioned them, I did not know them yet). $\endgroup$ – Lennart Jan 27 '12 at 18:50

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