An engineer would like to read further in details the above mentioned topics, I understand those are two separate topics on their own. But I was looking for a book treating both subjects with good transition. Appreciated.

In my work, I often model the physics of processes. I use the variational (weak relations) method to derive the plurality of numerical solutions to systems of nonlinear and coupled PDEs. During the courses I had taken, there was no rigorous treatment of the underlying mathematics. I would like to grasp how to arrive at generalizations, analysis of functional spaces, linear operators, particularly relations like green identities, or operators and tensor calculus.

That is why I asked for some book that starts with real then ends up with functional analysis with demonstration of the foundations and a presentation of the motivations throughout.

  • $\begingroup$ Honestly, I'm not sure whether you're going to be able to find a book that is that broad and also accessible. There are plenty of well-known but heavy texts that cover both subjects, like "Papa Rudin", but I'm not sure you will really get a better understanding with such a text than you would with a good RA text followed by a good FA text. As for an RA text I would recommend Royden/Fitzpatrick. $\endgroup$ – Ian Mar 28 '15 at 4:50
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    $\begingroup$ Functional analysis is so dependent upon a thorough understanding of real analysis that I think no coherent (and relatively succinct) text on both could exist. The closest thing I know to what you're looking for is Kreyszig's book. The real analysis treatment is a bit weak though. $\endgroup$ – Cameron Williams Mar 28 '15 at 4:55
  • $\begingroup$ There are very many questions of this sort on math.stackexchange already. Have you looked at them and found them dissatisfactory? If so, what were they missing that you would like instead? Have you looked at any book so far? $\endgroup$ – davidlowryduda Mar 28 '15 at 5:04
  • $\begingroup$ I've looked Lieb and Loss, it addresses functional analysis mainly but skims over real analysis... it takes many short cuts though. If i can find one that would put me on the road to learning the foundations of functional/real analysis in a modicum amount of time. Not trying to become a math professor yet $\endgroup$ – Sam Gomari Mar 28 '15 at 5:14
  • $\begingroup$ I do not recommend Lieb and Loss as a starter book in analysis as it is fairly sophisticated and doesn't give enough practice with basic analysis. What is the purpose of the study? The Stein and Shakarchi series (I, III and IV) might be good. They use Fourier analysis to motivate many ideas in real analysis, then cover measure theory and functional analysis in the first, third and fourth volumes respectively. $\endgroup$ – snar Mar 28 '15 at 6:31

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