Mathematical analysis - text book recommendation sought A rather subjective question, I admit, but I'm looking for a recommendation for a textbook to help me improve my understanding of mathematical analysis.
I come from a computing background, with a University level degree. My high-school mathematics was all focused on mechanics/physics rather than pure maths and statistics.  I've followed formal undergraduate courses on signal processing, and informal postgraduate lecture series subsequently - for personal interest, not credit.  I feel confident that I have the skills to digest any well-written undergraduate or masters-level text.
I recognise that I am (relatively) weak with respect to analysis when I read the Wikipedia pages for subjects such as:


*

*http://en.wikipedia.org/wiki/Non-analytic_smooth_function

*http://en.wikipedia.org/wiki/Distribution_%28mathematics%29

*http://en.wikipedia.org/wiki/Mollifier

*http://en.wikipedia.org/wiki/Holomorphic_function 

*etc. etc.


I am interested to bolster my understanding of the principles of mathematical analysis and to go on from this to improve my understanding of distributions and how they relate to both analytic and non-analytic functions.  While I recognise the value of proofs, and I'm not looking for a reference book from which to crib formulae, my principal interest is in the practical application of theory rather than its elegant abstract justification.  For this reason, I'm drawn more to presentations with intuitive over formal justifications for theorems.
Suggestions?
 A: At the very least you should start with Rudin's Principles of Mathematical Analysis. It is abstract, it is formal, but with many things mathematical, the devil is in the details. "Intuition" can easily lead you astray if you are not prepared to back it up with solid understanding. (Intuition is also usually developed after doing a lot of homework problems ;p.)
Then, since you are interested in developing "pure" intuition, I would suggest reading through Counterexamples in Analysis by Gelbaum and Olmsted. Analysis is largely naively intuitive... except where it is not. And if you want to come to an understanding of mathematical analysis by acquiring intuition, it is better to be forewarned about where your intuitions may go wrong. 
A: For distributions and mollifiers you might want to check out "Functional Analysis and Applications" by Kesavan.
A: I do not know your background, you are now student or graduated, but if you want to understand and have solid background on Analysis, as Willie Wong said, Rudin's Principle of Mathematical Analysis is the best one. However I think you should have a look at Mathematical Analysis by Zorich, published by Springer in the Universitext serie. This two volume covers from the basis to quite hard topics.
A: I would say Rudin's is arguably for readers of more mathematical maturity. By mathematical maturity I mean at least a sense of the overall picture of pure mathematics.
If your background lacks this feature, I would suggest you start with Apostol's Calculus. Do not be misled by the title. The book can give you both the introductory theory and techniques of Calculus, where the theory part can exactly give you a feeling of doing analysis manually. 
As another choice, I recommend Bartle's Real Analysis (all the books mentioned here can be easily found on the Internet simply be typing the keywords left here). Bartle's this book gives a rigorous treatment of introductory analysis (without measure theory). To me, it is also a great choice.
Then you may proceed to measure theory and other advanced parts of analysis. 
Be warned: To go into pure mathematics it is necessary to know math. logic and set theory to a satisfactory extent.  
A: Even "baby Rudin" seems a bit intimidating at this level. I honestly don't have a good suggestion for "first year analysis", which in many universities is called "advanced calculus". I used The Way of Analysis by Strichartz, but I wasn't thrilled by it.
After that I would wholeheartedly recommend Real Analysis by Royden and Fitzpatrick. If you need complex analysis, use a separate book. (Again I don't have a good example here, I used Gamelin but really didn't care for it.)
