I find it very difficult to understand analysis, because I can't find a way to learn it geometrically. To make my point clearer, let me take calculus as the example in contrast. I find calculus very geometric and I'm very comfortable with it, to name a few famous examples:

  1. Newton-Leibniz theorem: a strip under the graph of a (continuous)function, divided by its width, will approach the value of the height as width goes to 0;

  2. L'Hospital's Rule: if two functions are approaching 0 at the same point, as we approach that point, the ratio of the heights of the two graphs will be the same as the ratio of the slopes, because $\text{height}\approx \text{slope}\times\Delta x$, and $\Delta x$ is the same for both graphs.

I can keep going, and in fact, most of the important theorems in calculus have such geometric intepretations. I'm aware that there are technicalities that cannot be captureed by the above geometric arguments, but I believe they are still very to the point and I feel I can never forget them.

It seems to be another story for analysis(especially real analysis). Although occasionally there are important content that can be understood geometrically, more often I encounter theorems and concepts that after quite some struggling still no firm geometric intuition can be developed. A lot of times some partial intuition can be developed, but usually trusting these partial intuitions will quickly get myself into errors. I can follow most of the analysis textbooks, i.e. all their logical deductions, proving theorems etc. but the problem is, when I can't form pictures, I can't find positions in my mind for these knowledge, so although I have read a considerable amount of analysis(and done quite some exercises), not much can be "triggered" from my mind when I see a analysis-related problem , say, 4 months after I stop reading an analysis book.

In fact, I post the question here only after reading Poincare's Intuition and Logic in Mathematics, before reading this I assumed everyone thinks in the geometric way so if I can't understand analysis I probably should just try harder. It's quite a headache to me because nothing else(calculus, linear algebra, modern differential geometry etc.) has caused this much of discomfort. Now I'm wondering if I've learnt analysis in a wrong way, any comments or suggestions?

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    $\begingroup$ Intuition, logically may be somewhat different to your aim of intuition. Intuitionistic Logic is a mathematical effort to describe mathematics somehow in an algorithmic way. Mathematical Analysis simply is a generation of calculus, but by mathematical approach. I think Analysis is closer to Physics than geometry. $\endgroup$ – RSh Sep 12 '13 at 15:24
  • $\begingroup$ @I'mtoo: Thanks for pointing that out, I didn't know such thing as "Intuitionistic Logic" exists, may be I should change the title to avoid confusion. $\endgroup$ – Jia Yiyang Sep 12 '13 at 15:40
  • $\begingroup$ Does this answer your question? Visual book of real analysis $\endgroup$ – ugro Jul 22 at 8:38

I am not sure I agree with your statement that in real analysis it is hard to draw pictures. Most of the analysis on metric spaces can be visualized and pictures can be drawn. I learned most of the proofs in Baby Rudin by drawing pictures.

Concepts of open sets, closed sets, limits, continuity, compactness, convergence of sequences, definition of Riemann integration, etc can all be understood with help of pictures.

Once you go to function spaces, things get weird, but even then, in Hilbert spaces you can basically use all of your geometric intuition.

  • $\begingroup$ My (possibly naive) intuition often fails me and sometimes causes more confusion than clarification, say, the $\epsilon-\delta$ definition of continuity of a function $f$, it is somewhat in accord with a geometric continuous line; another equivalent definition is $f^{-1}$ maps every closed set to closed set, for this one my intuition would be since closed set and open set are topologically distinct, if we want to map a closed set to an open set, a "tearing" must happen, and a "tearing" can't be continuous, but why this condition for continuity is imposed on $f^{-1}$ not $f$? Of course if we $\endgroup$ – Jia Yiyang Sep 12 '13 at 15:34
  • $\begingroup$ work out the deductions, we'll see the former is equivalent to $\epsilon-\delta$, but my intuition is somehow still asking why not "$f$ maps every closed set to closed set" is the one that charaterize the continuous behavior? $\endgroup$ – Jia Yiyang Sep 12 '13 at 15:38
  • $\begingroup$ @JiaYiyang a set can be closed and open. $\endgroup$ – user12802 Sep 12 '13 at 15:46
  • $\begingroup$ @Andrew: Ok I was having intervals on real line in mind, anyway that's the closest thing to an intuition I could get from the "closed set to closed set" definition. $\endgroup$ – Jia Yiyang Sep 12 '13 at 15:50

Rather than a comment, I will post this as an answer which I hope will be beneficial. Charles Pugh wrote what I think is one of the most readable and excellent text for learning real analysis. He rightfully emphasizes the importance of pictures in getting it. So this book may be appealing to you.


I have used it extensively with pencil and paper at hand. Once you just start drawing the pictures, you will get used to thinking that way. After all it is REAL analysis, so there is always a drawable picture. In fact, to substantiate your understanding and convince yourself that you really get it, you would want to draw a picture. Good luck.

  • $\begingroup$ Thanks for the reference, I'll take a look when I get the time. $\endgroup$ – Jia Yiyang Sep 12 '13 at 16:11

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