Lemma (mathematics)
Encyclopedia
In mathematics
, a lemma (plural lemmata or lemmas from the Greek
λήμμα, "lemma" meaning "anything which is received, such as a gift, profit, or a bribe") is a proven proposition which is used as a stepping stone to a larger result rather than as a statement in-and-of itself. There is no formal distinction between a lemma and a theorem
, only one of usage and convention – see Theorem#Terminology.
A good stepping stone leads to many others, so some of the most powerful results in mathematics are known as lemmas, such as Bézout's lemma, Urysohn's lemma
, Dehn's lemma
, Fatou's lemma
, Gauss's lemma
, Nakayama's lemma, Poincaré's lemma, Riesz's lemma
, Schwarz's lemma, Itō's lemma
and Zorn's lemma
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a lemma (plural lemmata or lemmas from the Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
λήμμα, "lemma" meaning "anything which is received, such as a gift, profit, or a bribe") is a proven proposition which is used as a stepping stone to a larger result rather than as a statement in-and-of itself. There is no formal distinction between a lemma and a theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
, only one of usage and convention – see Theorem#Terminology.
A good stepping stone leads to many others, so some of the most powerful results in mathematics are known as lemmas, such as Bézout's lemma, Urysohn's lemma
Urysohn's lemma
In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a function....
, Dehn's lemma
Dehn's lemma
In mathematics Dehn's lemmaasserts that a piecewise-linear map of a disk into a 3-manifold, with the map's singularity set in the disc's interior, implies the existence of another piecewise-linear map of the disc which is an embedding and is identical to the original on the boundary of the...
, Fatou's lemma
Fatou's lemma
In mathematics, Fatou's lemma establishes an inequality relating the integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions...
, Gauss's lemma
Gauss's lemma
Gauss's lemma can mean any of several lemmas named after Carl Friedrich Gauss:* Gauss's lemma * Gauss's lemma * Gauss's lemma...
, Nakayama's lemma, Poincaré's lemma, Riesz's lemma
Riesz's lemma
Riesz's lemma is a lemma in functional analysis. It specifies conditions which guarantee that a subspace in a normed linear space is dense.- The result :...
, Schwarz's lemma, Itō's lemma
Ito's lemma
In mathematics, Itō's lemma is used in Itō stochastic calculus to find the differential of a function of a particular type of stochastic process. It is named after its discoverer, Kiyoshi Itō...
and Zorn's lemma
Zorn's lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory that states:Suppose a partially ordered set P has the property that every chain has an upper bound in P...
.
External links
- Doron ZeilbergerDoron ZeilbergerDoron Zeilberger is an Israeli mathematician, known for his work in combinatorics.He is a Board of Governors Professor of Mathematics at Rutgers University...
, Opinion 82: A Good Lemma is Worth a Thousand Theorems