4
$\begingroup$

When studying Analysis (Real, $\mathbb R^n$, Metric ...) it's common to see similar (or even the same) theorem, sometimes with the same (or similar) proof. I was wondering if there is any book out there that tries to present "Analysis" with results from a more general spaces to more specific spaces. For example, if a result is valid for every complete metric space, then it's presented and proved with this generic view. While, if the result is specific to $\mathbb R$, then the result is shown for that case.

I understand that this might be perhaps too much to ask, but you never know. Maybe someone wrote such book already.

Just to make clear where this comes from. I'm writing some notes on courses I've taken as a graduate student, and I was trying to organize my notes from "more generic to more specific". Hence, I started with stuff like Topological Spaces, Metric Spaces, and moved to Complex and Euclidean Spaces. I realized that many of the results I proved for Euclidean Spaces were pretty much the same in more general spaces, and I was repeating a bunch of my theorems, with slight changes.

$\endgroup$
4
  • $\begingroup$ In real analysis we study the $(\Bbb{R}, \|•\|_2) $ which is also a metric space, topological space, topological vector space. $\Bbb{R^n}$ is also a Banach space(Which Norm doesn't matter! ) So for a general overview you can study topological vector space. But you have to know a lot of stuff. Vector space, metric space, normed space, topological space etc. $\endgroup$
    – S. G
    Jan 14 at 19:23
  • 1
    $\begingroup$ My first suggestion would be to look for the relevant Bourbaki volumes. Although it might not be such a great idea… $\endgroup$
    – Mindlack
    Jan 14 at 19:24
  • $\begingroup$ Why Bourbaki would not be a great idea? I have no problem with abstraction :) $\endgroup$ Jan 14 at 20:52
  • 1
    $\begingroup$ @S.G, I've seen most of these topics (some more deeply). I even went back as much as Category Theory in order to try to get a more general view of things ehehe. $\endgroup$ Jan 14 at 20:53
2
$\begingroup$

I think you would be strongly interested in the book "Mathematical Analysis II" from Vladimir Zorich.

$\endgroup$
0
2
$\begingroup$

The three volumes Analysis I, Analysis II and Analysis III by Amann and Escher are very general. I can't speak to your specific example but they try to present the material, starting from practically nothing, as a professional mathematician might view it, so it's quite general.

See the Amazon reviews for evidence of A&E's generality.

The book Real and Functional Analysis by Lang is similar, though it's arguably even tougher because it's only about half as long as the three volumes by Amann and Escher, and it goes quite a bit further, specifically into functional analysis.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.