Lemma (mathematics)

In mathematics, informal logic and argument mapping, a lemma (plural lemmas or lemmata) is a generally minor, proven proposition which is used as a stepping stone to a larger result. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem".[1][2][3] 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.[4] The word "lemma" derives from the Ancient Greek λῆμμα ("anything which is received"[3], such as a gift, profit, or a bribe).

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 theorem  – a step in the direction of proof[4] – or a short theorem appearing at an intermediate stage in a proof.[5]

Well-known lemmas

A good stepping stone can lead to many others. Some powerful results in mathematics are known as lemmas. These include, among others:

• Bézout'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
• 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

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.

See also

• Corollary
• Co-premise
• Fundamental lemma
• Inference objection
• List of lemmas
• Objection
• Porism
• Theorem
• Theorem terminology

References

1. "The Definitive Glossary of Higher Mathematical Jargon — Lemma". Math Vault. 2019-08-01. Retrieved 2019-11-28.
2. Higham, Nicholas J. (1998). Handbook of Writing for the Mathematical Sciences. Society for Industrial and Applied Mathematics. pp. 16. ISBN 0-89871-420-6.
3. "Definition of lemma | Dictionary.com". www.dictionary.com. Retrieved 2019-11-28.
4. Richeson, Dave (2008-09-23). "What is the difference between a theorem, a lemma, and a corollary?". David Richeson: Division by Zero. Retrieved 2019-11-28.
5. Wolfram, Stephen (2002). A New Kind of Science. Wolfram Media, Inc. p. 1176. ISBN 1-57955-008-8.

External links

• Doron Zeilberger, Opinion 82: A Good Lemma is Worth a Thousand Theorems

This article incorporates material from Lemma on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.