Lemma (mathematics)

Etymology

From the Ancient Greek λῆμμα, (perfect passive εἴλημμαι) 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 (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.

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 (any of several named after Carl Friedrich Gauss)
• 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

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.

Notes

References

References

1. [https://www.merriam-webster.com/dictionary/lemma.] "Lemma." Merriam-Webster.com Dictionary, Merriam-Webster.
2. Loewen, Nathan R. B. ''Beyond the Problem of Evil.'' Lexington Books. March 12, 2018. {{ISBN. 9781498555739 p. 47
3. Higham, Nicholas J.. (1998). "Handbook of Writing for the Mathematical Sciences". [[Society for Industrial and Applied Mathematics]].
4. "Definition of lemma {{!}} Dictionary.com".
5. Richeson, Dave. (2008-09-23). "What is the difference between a theorem, a lemma, and a corollary?".
6. "Oxford English Dictionary". Oxford University Press.