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I have made a big mistake by signing up to an advanced functional analysis class, when I do not have the background. This is my first analysis course ever, and aside from that my math background is very light (one semester calculus, two semesters stats, one semester game theory). Alas, the die has been cast and now I am trying to keep my head above water and survive(pass) the course. I have a midterm in about 20 days from the date of this publication. I would like some opinion on how I could muster a semi-decent-not-really grasp of what is going on to get half the marks(just enough to pass). We have basically gone through Metric, Banach, and Hilbert spaces(their topological properties and key theorems to analyse them) and some applications. I have been told that the exam will be to apply the properties of these to economic problems (I am an economics major). Any advice on how I could approach my studying to be efficient would be greatly appreciated. Thank you!

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    $\begingroup$ My condolences. You need massive support if you are to do this. I would come clean and talk to the professor and administration to back out (uncast the die). Otherwise, you need a tutor and a substantial amount of time to find and repair the holes in your background. (Let the professor know about this: (s)he might help.) Get a past exam, and have you and the tutor identify the weak points in your knowledge as the two of you walk through a solution. Given the time constraint, I am confident that you will not succeed at this by yourself. Gerhard "Asking For Help Is Good" Paseman, 2015.10.21 $\endgroup$ – Gerhard Paseman Oct 21 '15 at 16:44
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    $\begingroup$ Here is another idea: pretend you are teaching someone the material, and write down all of the notes, summaries, mnemonics for the concepts that you think will be on the midterm. Spend two days on this. Find the holes and then go to the professor and tell them this is the first stage of your preparation, and ask what would (s)he would prioritize. Likely you will impress them with your efforts to meet the course halfway, and they might tell you the key insights you need to cover the things you know you don't understand. Gerhard "Make Your Ignorance Specifically Visible" Paseman, 2015.10.21 $\endgroup$ – Gerhard Paseman Oct 21 '15 at 16:51
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    $\begingroup$ I fear that you cannot solve this problem given your constraints... As soon as possible, go to the instructor and explain your situation. (That "functional analysis" would often be a second-year graduate mathematics course in the U.S., for people who had (or tested out of) four semesters of calculus, some "advanced calculus", maybe a year or_two of "real analysis"... not to mention other math classes that help "maturity".) $\endgroup$ – paul garrett Oct 21 '15 at 21:11
  • $\begingroup$ This question is somewhat at the edge of what is on-topic here, you might try the site Mathematics with a similar question. Yet please do not post the exact same question there. In particular, if you post there cut down on the personal context or mention it towards the end: rather ask for material to get a quick grasp of the material you mentioned. Good luck! $\endgroup$ – quid Oct 23 '15 at 9:40
  • $\begingroup$ You might get some fast traction from "Topology Without Tears" by Sidney A. Morris. It's free(!) and I found it extremely useful in a not too dissimilar position as yours. Also you're probably building up to Martingales, so any beginner's tutorial to that would be useful $\endgroup$ – user2121 Oct 31 '15 at 2:49

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