Lemma (mathematics)

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In mathematics and other fields,Template:Efn a lemma (Template:Plural form: lemmas or lemmata) is a generally minor, proven proposition which is used to prove a larger statement. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem".<ref>Template:Cite book</ref><ref name=":0">Template:Cite web</ref> In many cases, a lemma derives its importance from the theorem it aims to prove; however, a lemma can also turn out to be more important than originally thought.<ref name=":1">Template:Cite web</ref>

Etymology

From the Ancient Greek λῆμμα, (perfect passive εἴλημμαι) something received or taken. Thus something taken for granted in an argument.<ref name="OED">Template:Cite web</ref>

Comparison with theorem

There is no formal distinction between a lemma and a theorem, only one of intention (see Theorem terminology). However, a lemma can be considered a minor result whose sole purpose is to help prove a more substantial theorem – a step in the direction of proof.<ref name=":1"/>

Well-known lemmas

Some powerful results in mathematics are known as lemmas, first named for their originally minor purpose. These include, among others:

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Bézout's lemma
Burnside's lemma
Dehn's lemma
Euclid's lemma
Farkas' lemma
Fatou's lemma
Gauss's lemma (any of several named after Carl Friedrich Gauss)
Greendlinger's lemma
Itô's lemma
Jordan's lemma
Lovász local lemma
Nakayama's lemma
Poincaré's lemma
Riesz's lemma
Schur's lemma
Schwarz's lemma
Sperner's lemma
Urysohn's lemma
Vitali covering lemma
Yoneda's lemma
Zorn's lemma

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While these results originally seemed too simple or too technical to warrant independent interest, they have eventually turned out to be central to the theories in which they occur.