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An engineer with a master's degree, I would like to improve my level of mathematical rigor. For example, I would like to come to grasp

  • what measure theory has to do with the Fourier series and
  • what group theory has to do with the eigenvalue,

among others. I would like to understand why Lebesgue integration is important (the idea seems sort of trivial to me, which naturally means that I don't yet really get it), why the formality of Weierstrassian tests is preferable to the mere evasion of the problem by, say, the extraction of Hadamard's finite part, and so on.

To this end, I have been reading Georgi E. Shilov's Elementary Real and Complex Analysis, which I like; but at the rate Shilov is going it would take like 10,000 pages to advance as far as the Fourier series. Therefore, I have looked instead on Amazon and Google Books for the book Mathematical Rigor for Engineers.

Unfortunately, no one seems to have written that book.

What are my options?

(To clarify: I do not mean to ask the opinion-based question, "What should I do?" Rather, I mean to ask the experience-based question, "What are my options?" Maybe the only answer is that what I seek resembles Financial Analysis for Kindergarteners or Transoceanic Logistics for Retail Clerks, but since I happen to possess an applicationist's acquaintance with mathematical productions like the Fourier Series and the eigenvalue, not to mention Hadamard's finite part, I thought that I would ask.)

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  • $\begingroup$ Have you had a serious linear algebra course, i.e. following Friedberg Insel and Spence's Linear Algebra/Hoffman & Kunze/Axler/something else at that rough level? $\endgroup$ – Batman Jun 13 '17 at 1:53
  • $\begingroup$ @Batman: stand by.... My linear algebra preceptor is Joel N. Franklin, so let me go to look up Friedberg Insel and friends now.... $\endgroup$ – thb Jun 13 '17 at 1:58
  • $\begingroup$ I don't think it's easy to answer a general question like this. I think it would make more sense to ask about individual topics. For example, "I have X background knowledge in analysis. I would like to learn about mathematical subject Y in a mathematically rigorous way, with an emphasis on the areas of the theory that are important in applied field Z. How can I do this in the most direct way possible?" Depending on the specific choice of X, Y and Z, you will get answers that are very different. If you are interested in many fields Y, then it would be advisable to take the same learning path... $\endgroup$ – user49640 Jun 13 '17 at 2:05
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    $\begingroup$ ...that math students do, including a broad initial course in analysis. In fact, because you have mentioned that you would like to learn measure theory, I think it's almost unavoidable that you will need a good background, in analysis and linear algebra at least, similar to that of math undergraduates. $\endgroup$ – user49640 Jun 13 '17 at 2:06
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    $\begingroup$ @user49640: I see. In general, it is very hard to know where to begin. The very names mathematicians use for their subjects, like analysis, tend carry unrelated semantics to engineers. The thing is this: the more one learns, the more favorably impressed one becomes with the character of the typical professional mathematician. I don't believe that most mathematicians are deliberately trying to be obscure; and I do believe Felix Klein when he says that mathematicians wish to be as intelligible to engineers as possible. But there is this gap. I would like to bridge it. $\endgroup$ – thb Jun 13 '17 at 2:13
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I think that what you're really looking for is Representation Theory. Now the problem is that it can be quite advanced for an engineer trying to approach this subject from a mathematical point of view. But maybe it can be done from a more "physical" point of view.

I would suggest to you "Lectures on Linear Algebra" from Gelfand that if I rember correctly also adresses Fourier Series, and then the second volume of "Principles of Advanced Mathematical Physics Volume 2" of Richtmeyer which has in the first 10 chapter the basis of representation theory in a very comprehensible way. I'm not sure on this last one since it might be too advanced, but I would give it a try if I were you.

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