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Can someone recommend me an analysis book in which every (or at least most) theorem(s) are given using minimal premises ?

By this I mean that most analysis books in which I looked present give the theorems under premises that are a little bit to strong, for example they often require $f$ to be continuous where it would suffice for $f$ to be $R$-integrable or (in the case of second order derivatives) they require that $f$ is continuously differentiable in an interval around $a$ (if $a$ is the point in which the second order derivative is to be taken), where it would suffice to merely state that $a$ is an accumulation point for the "second order difference function" which in the limit to $a$ gives the second order derivative.

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Analysis is a huge subject - which parts are you looking for? What is your background/preliminaries? – AD. Oct 20 '11 at 10:48
Walter Rudin wrote some excellent books in analysis. – AD. Oct 20 '11 at 10:49
@AD. My preliminaries are fairly low, mening I'm looking for a book that treats (rigorously) the analysis of functions of one and several real variables (if its analysis in Banach spaces its also ok, as long as the connections to the analysis of reals variables are made) – resu Oct 22 '11 at 9:07
up vote 4 down vote accepted

I think that rather than looking for such a book, you should have a lot of analysis books on hand (purchase, borrow from library, etc.). Then, when you come to a certain topic in your primary book/text, you can look up that topic's treatment in other books. You'll often find that Author A gives a more refined version than Author B of Theorem X, but then Author B gives a more refined version than Author A of Theorem Y. Also, the push towards having the least hypotheses can quickly lead to technical issues beyond the scope of your current interests. For example, if you really want something that involves fairly minimal hypotheses, take a look at Krishna M. Garg's book Theory of Differentiation: A Unified Theory of Differentiation Via New Derivate Theorems and New Derivatives. Here, instead of a hypothesis of continuity or even of semicontinuity on an interval, you'll find a hypothesis that may be something like "unilaterally quasi-semicontinuous on a co-countable set". However, my guess is that Garg's book is not what you're looking for. That said, I suggest looking at Andrew Gleason's Fundamentals of Abstract Analysis.

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For the people, who -- like me -- do not have access to a copy of Garg's book, here's a sample from google books: the formulation of Mean value theorem on p.166 – Martin Sleziak Oct 20 '11 at 15:59
@Dave L: Renfro Indeed,Garg's book wasn't what I had in mind, but nonetheless it was interesting to find that such a book exists. And Gleason's prett much is what I was looking for, great answer, thanks! But I'm not going to mark your answer as the correct one yet in the hope the some other books will be recommended, because seeing that there are very interesting books out there I don't think it would be wise to "close" this question yet, by accepting an answer. – resu Oct 22 '11 at 9:50
ok, since no further books have been listed, I'm going to accept the answer now. – resu Oct 27 '11 at 9:15

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