Lemma (mathematics)
In mathematics and other fields, a lemma 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". 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.
Etymology
From the Ancient Greek λῆμμα, something received or taken. Thus something taken for granted in an argument.Comparison with theorem
There is no formal distinction between a lemma and a theorem, only one of intention. 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.Well-known lemmas
Some powerful results in mathematics are known as lemmas, first named for their originally minor purpose. These include, among others:- Bézout's lemma
- Burnside's lemma
- Dehn's lemma
- Euclid's lemma
- Farkas' lemma
- Fatou's lemma
- Gauss's lemma
- Greendlinger's lemma
- Itô's lemma
- Jordan's lemma
- Lovász local lemma
- Nakayama's lemma
- Noether normalization 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
- Zariski's lemma
- Zorn's lemma