How to use mathematical concepts and statements

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How to use mathematical concepts and statements

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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 Baire category theorem]]

[[How to use Bayes's theorem]]

[[How to use the Bolzano-Weierstrass theorem]] [cut] Quick description || The Bolzano-Weierstrass theorem asserts that every bounded sequence of real numbers has a convergent subsequence. More generally, it states that if $X$ is a closed bounded subset of $\mathbb{R}^n$ then every sequence in $X$ has a subsequence that converges to a point in $X$. 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. [/cut]

[[How to use the Cauchy-Schwarz inequality]]

[[How to use the central limit theorem]]

[[How to use the classification of finite simple groups]]

[[How to use cohomology]]

[[How to use compactness]]   [cut] 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.[/cut]

[[How to use the continuum hypothesis]]  [cut] Quick description || Cantor's continuum hypothesis is perhaps the most famous example of a mathematical statement that turned out to be independent of the [[w:Zermelo-Fraenkel_set_theory|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. [/cut]

[[How to use correlation inequalities]]

[[How to use duality]]

[[How to use entropy]]

[[How to use exact sequences]]

[[How to use fixed point theorems]]

[[Fourier transforms front page|How to use the Fourier transform]]

[[How to use generating functions]]

[[How to use group actions]]

[[How to use the Hahn-Banach theorem]]

[[How to use the inclusion-exclusion principle]]

[[How to use Janson's inequality]]

[[How to use the Lov\'asz local lemma]]

[[How to use martingales]]

[[How to use the max-flow-min-cut theorem]]

[[How to use the mean value theorem]]

[[How to use ordinals]]

[[Decompose your ring using idempotents | How to use the Peirce decomposition ]][cut] Quick description || If $R$ is a ring and $e \in R$ is an idempotent, then the Peirce decomposition of $R$
is the decomposition $R = e R e \oplus e R (1-e) \oplus (1-e) R e \oplus
(1-e) R(1-e).$[/cut]

[[How to use the pigeonhole principle]]

[[How to use the Riemann-Roch theorem]]

[[How to use spectral gaps]]

[[How to use spectral sequences]]

[[How to use Szemer\'edi's regularity lemma]]

[[How to use Talagrand's inequality]]

[[How to use tensor products]]
*[[How to use tensor products and evaluation maps in representation theory]]

[[How to use ultrafilters]] [cut] Quick description || An ''ultrafilter'' on a set $X$ is a collection $\mathcal{U}$ of subsets of $X$ with the following properties: (i) $\emptyset\notin\mathcal{U}$; (ii) $\mathcal{U}$ is closed under finite intersections; (iii) if $A\in\mathcal{U}$ and $A\subset B$ then $B\in\mathcal{U}$; (iv) for every $A\subset X$, either $A$ or $X\setminus A$ belongs to $\mathcal{U}$. A trivial example of an ultrafilter is the collection of all sets containing some fixed element $x$ of $X$. 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. [/cut]

[[How to use Zorn's lemma]]    [cut] 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.[/cut]<comment thread="118" />

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How to use mathematical concepts and statements

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