Anybody who has taken an undergraduate mathematics course, and certainly anybody who has taught undergraduate mathematics, will know that there is a huge difference between being familiar with a theorem and knowing how to use it. However, it is a widely adopted convention in textbooks and lectures to give a theorem and its proof and then to hope that the audience will somehow work out how it is applied. One way this is done is through the setting of exercises, and often the main difficulty in solving an exercise is spotting the appropriate theorem to use. Something similar can be true at the research level too: a problem that seems hard to one mathematician may well be easy to another who recognises that it is a consequence of a theorem that is designed to deal with exactly that difficulty.
This is a navigation page with a list of Tricki articles, each of which is entitled "How to use X" for some X. We give the titles and quick descriptions of the articles. (To see the latter, click on the words "Quick description".) Probably it will at some point get too big to be convenient to use. At that point, this page will become a front page with links to more specialized "How to use" pages.
How to use the Bolzano-Weierstrass theorem Quick description ( The Bolzano-Weierstrass theorem asserts that every bounded sequence of real numbers has a convergent subsequence. More generally, it states that if is a closed bounded subset of then every sequence in has a subsequence that converges to a point in . This article is not so much about the statement, or its proof, but about how to use it in applications. As explained in the article, there are certain signs to look out for: if you come across one of these signs then the Bolzano-Weierstrass theorem may well be helpful. )
How to use compactness Quick description ( Compactness is an all-pervasive concept in mathematics. But it is also a tool that can be used for solving problems. This article briefly explains some typical uses and gives links to other more detailed articles.)
How to use the continuum hypothesis Quick description ( Cantor's continuum hypothesis is perhaps the most famous example of a mathematical statement that turned out to be independent of the Zermelo-Fraenkel axioms. What is less well known is that the continuum hypothesis is a useful tool for solving certain sorts of problems in analysis. This article gives a few examples of its use. )
How to use the Peirce decomposition Quick description ( If is a ring and is an idempotent, then the Peirce decomposition of is the decomposition )
How to use ultrafilters Quick description ( An ultrafilter on a set is a collection of subsets of with the following properties: (i) ; (ii) is closed under finite intersections; (iii) if and then ; (iv) for every , either or belongs to . A trivial example of an ultrafilter is the collection of all sets containing some fixed element of . Such ultrafilters are called principal. It is not trivial that there are any non-principal ultrafilters, but this can be proved using Zorn's lemma. Here we explain various ways of using non-principal ultrafilters in analysis and infinitary combinatorics. )
How to use Zorn's lemma Quick description ( If you are building a mathematical object in stages and find that (i) you have not finished even after infinitely many stages, and (ii) there seems to be nothing to stop you continuing to build, then Zorn's lemma may well be able to help you.)♦