I need a dictionary for finding definitions, theorems to use in proofs in mathematical analysis. What can you recommend about it? I need something that is self-contained. While studying, I need to check a lot of definitions and theorems, and all of them are not in the same book, and in the net it is not much satisfactory. My textbook is not self-contained either
Thanks in advance!
On
Wikipedia is pretty useful, I find. There are sometimes errors, but not for the elementary stuff, mostly. I'd suggest an alternative, though: when you need a definition, look it up somewhere, and make a note, say "Upper semi-continuous", and put a check mark next to it. Each time you look it up, put another check mark. When you get to three, memorize it.
There's a reason that folks define certain things -- the choices aren't handed down on a stone tablet from Weierstrass or anything. They give certain things names because those things keep coming up. So pretty much any definition you encounter might well be worth memorizing.
The same goes for theorems, but moreso. You should say things like "Hmmm... I've got a limit outside an integral. When can I pull it inside?" and a couple of useful theorems (dominated convergence, for instance) should jump to mind. Every time you use a theorem, put a check mark next to it. When you get three check marks, memorize it. (And you might want to memorize the proof as well.)
As an alternative, you can just write down every definition and theorem you encounter -- make a nice document in your favorite math typesetting language (I'd use LaTeX, but others have other preferences), and start adding stuff to it. That way, you'll have read, written, and re-read every theorem, which is not a bad place to start. When you're done, you'll have an excellent "dictionary." :)
I have not heard of such an analysis dictionary, and I would suggest that there is no need for that. In fact, not having such a dictionary can prove to be a big motivation to keep definitions and theorems in one's mind. You look at whatever you have to look at to recall a theorem five six seven... times, until you get so bored and tired of searching for stuff. Then it's way efficient to keep those in your memory. At least that's what I think.
As for a more objective answer, note that mathematics, or analysis by itself, is quite a large subject. Consequently everything is connected (more or less), so that a self-contained text always denotes some trade-offs. What is more, most of the time there are many ways to approach a subject, e.g. take the definition of a closed subset of a metric space. I first learned the sequential definition (so "being closed" is "being closed under convergence"). But you might as well start with the topological definition (complement is open), or the definition via the closure operator (if its closure is itself). So, if you have a fixed textbook, chances are such a dictionary won't be more easier to use than using a few textbooks at the same time (not that using a few textbooks at the same time is quite fun, at least in terms of how much they can improve your conceptual flexibility). The notation, the approach etc. will probably be equally diverse.
Another suggestion would be to take out your lecture notes (if you have) and your textbooks and make a list of all results, definitions, "canonical" examples and remarks in the order you see fit. I do this usually when I'm studying for some exam. I leave the proofs out, and then try to (re)construct the proofs for the most important ones. This way you will obtain a very individualized narrative. For instance I like to write everything down in a very formal language, and sometimes that provides one some bonus challenge. Needless to say, creating your own dictionary could take a lot of time if you're not used to it.