List of lemmas
Encyclopedia
This following is a list of lemmas
(or, "lemmata", i.e. minor theorem
s, or sometimes intermediate technical results factored out of proofs). See also list of axioms, list of theorems and list of conjectures.
Lemma (mathematics)
In mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than as a statement in-and-of itself...
(or, "lemmata", i.e. minor 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...
s, or sometimes intermediate technical results factored out of proofs). See also list of axioms, list of theorems and list of conjectures.
A to E
- Abel's lemma (mathematical series)
- Abhyankar's lemmaAbhyankar's lemmaIn mathematics, Abhyankar's lemma allows one to kill tame ramification by taking an extension of a base field....
(algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
) - Archimedes' lemmasBook of LemmasThe Book of Lemmas is a book attributed to Archimedes by Thābit ibn Qurra, though the authorship of the book is questionable. It consists of fifteen propositions on circles.-Translations:...
(euclidean geometryEuclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
) - Artin–Rees lemma (commutative algebraCommutative algebraCommutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
) named after Emil ArtinEmil ArtinEmil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...
and Elmer ReesElmer ReesElmer Gethin Rees, CBE, FRSE is a mathematician with publications in area ranging from topology, differential geometry, algebraic geometry, linear algebra and Morse theory to robotics... - Aubin–Lions lemma
- Barbalat's lemmaLyapunov stabilityVarious types of stability may be discussed for the solutions of differential equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Lyapunov...
(dynamical systemDynamical systemA dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s) - Basic perturbation lemma (computer scienceComputer scienceComputer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
, algebraAlgebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
) - Berge's lemmaBerge's lemmaIn graph theory, Berge's lemma states that a matching M in a graph G is maximum if and only if there is no augmenting path with M.It was proven by French mathematician Claude Berge in 1957.- Proof...
(graph theoryGraph theoryIn mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
) named after Claude BergeClaude BergeClaude Berge was a French mathematician, recognized as one of the modern founders of combinatorics and graph theory. He is particularly remembered for his famous conjectures on perfect graphs and for Berge's lemma, which states that a matching M in a graph G is maximum if and only if there is in... - Bézout's lemmaBézout's identityIn number theory, Bézout's identity for two integers a, b is an expressionwhere x and y are integers , such that d is a common divisor of a and b. Bézout's lemma states that such coefficients exist for every pair of nonzero integers...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Bhaskara's lemma (Diophantine equationDiophantine equationIn mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...
s) - Borel's lemmaBorel's lemmaIn mathematics, Borel's lemma is an important result about partial differential equations named after Émile Borel.Suppose U is an open set in the Euclidean space Rn, and suppose that f_0, f_1, ... is a sequence of smooth, complex-valued functions on U...
(partial differential equationPartial differential equationIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s) - Borel–Cantelli lemma (probability theoryProbability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
) - Bounding lemmas, of which there are several
- Bramble–Hilbert lemma (numerical analysisNumerical analysisNumerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
) - Brezis–Lions lemma
- Burnside's lemmaBurnside's lemmaBurnside's lemma, sometimes also called Burnside's counting theorem, the Cauchy-Frobenius lemma or the orbit-counting theorem, is a result in group theory which is often useful in taking account of symmetry when counting mathematical objects. Its various eponyms include William Burnside, George...
also known as the Cauchy–Frobenius lemma (group theoryGroup theoryIn mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
) - Céa's lemmaCéa's lemmaCéa's lemma is a lemma in mathematics. It is an important tool for proving error estimates for the finite element method applied to elliptic partial differential equations.-Lemma statement:Let V be a real Hilbert space with the norm \|\cdot\|...
(numerical analysisNumerical analysisNumerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
) - Closed map lemmaOpen and closed mapsIn topology, an open map is a function between two topological spaces which maps open sets to open sets. That is, a function f : X → Y is open if for any open set U in X, the image f is open in Y. Likewise, a closed map is a function which maps closed sets to closed sets...
(topologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
) - Closeness lemma (functionsFunction (mathematics)In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
) - Commutation lemmas, of which there are several
- Composition lemmas, of which there are several
- Cotlar–Stein lemmaCotlar–Stein lemmaIn mathematics, in the field of functional analysis, the Cotlar–Stein almost orthogonality lemma is named after mathematicians Mischa Cotlarand Elias Stein...
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Couchman's lemma (set theorySet theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
) - Counting lemmas, of which there are several
- Cousin's lemma (integralIntegralIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
s) - Covering lemmaCovering lemmaIn mathematics, under various anti-large cardinal assumptions, one can prove the existence of the canonical inner model, called the Core Model, that is, in a sense, maximal and approximates the structure of V...
(set theorySet theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
) - Craig interpolation lemmaCraig interpolationIn mathematical logic, Craig's interpolation theorem is a result about the relationship between different logical theories. Roughly stated, the theorem says that if a formula φ implies a formula ψ then there is a third formula ρ, called an interpolant, such that every nonlogical symbol...
(mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
) - Crossing lemmaCrossing numberCrossing number may refer to:*Crossing number of a knot is the minimal number of crossings in any knot diagram for the knot.**The average crossing number is a variant of crossing number obtained from a three-dimensional embedding of a knot by averaging over all two-dimensional...
(knot theoryKnot theoryIn topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
, graph theoryGraph theoryIn mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
) - Danielson–Lanczos lemma (Fourier transformFourier transformIn mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
s) - Davis–Figiel–Johnson–Pelczynski factorization lemma
- Dehn's lemmaDehn's lemmaIn 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...
(geometric topologyGeometric topologyIn mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
) - Delta lemmaDelta lemmaIn mathematics, a sunflower or Δ system is a collection of sets whose pairwise intersection is constant, and called the kernel.The Δ-lemma is a combinatorial set-theoretic tool used in proofs to impose an upper bound on the size of a collection of pairwise incompatible elements in a forcing...
(set theorySet theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
) - Deny–Lions lemma
- Diagonal lemmaDiagonal lemmaIn mathematical logic, the diagonal lemma or fixed point theorem establishes the existence of self-referential sentences in certain formal theories of the natural numbers -- specifically those theories that are strong enough to represent all computable functions...
(mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
) - Dickson's lemmaDickson's lemmaIn mathematics, Dickson's lemma is a finiteness statement applying to n-tuples of natural numbers. It is a simple fact from combinatorics, which has become attributed to the American algebraist L. E. Dickson...
(combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Dobrushin's lemma (point process theory)
- Dwork's lemma (number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Dynkin lemmaDynkin systemA Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set \Omega satisfying a set of axioms weaker than those of σ-algebra. Dynkin systems are sometimes referred to as λ-systems or d-system...
(set theorySet theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
) - Ehrling's lemmaEhrling's lemmaIn mathematics, Ehrling's lemma is a result concerning Banach spaces. It is often used in functional analysis to demonstrate the equivalence of certain norms on Sobolev spaces.-Statement of the lemma:...
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Ellis–Nakamura lemmaEllis–Nakamura lemmaIn mathematics, the Ellis–Numakura lemma states that if S is a non-empty semigroup with a topology such thatS is compact and the product is semi-continuous, then S has an idempotent element p, .-Applications:...
(topological semigroupTopological semigroupIn mathematics, a topological semigroup is a semigroup which is simultaneously a topological space, and whose semigroup operation is continuous.A topological group is a topological semigroup.- See also :* Strongly continuous semigroup* Analytic semigroup...
s) - Estimation lemmaEstimation lemmaIn mathematics, the estimation lemma gives an upper bound for a contour integral. If f is a complex-valued, continuous function on the contour \Gamma and if its absolute value |f| is bounded by a constant M for all z on \Gamma, then...
(contour integrals) - Euclid's lemmaEuclid's lemmaIn mathematics, Euclid's lemma is an important lemma regarding divisibility and prime numbers. In its simplest form, the lemma states that a prime number that divides a product of two integers must divide one of the two integers...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Expander mixing lemmaExpander mixing lemmaThe expander mixing lemma states that, for any two subsets S, T of a regular expander graph G, the number of edges between S and T is approximately what you would expect in a random d-regular graph, i.e. d |S| \cdot |T| / n.-Statement:...
(graph theoryGraph theoryIn mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
) - Expansion lemmas, of which there are several
F to J
- Factorization lemmaFactorization lemmaIn measure theory, the factorization lemma allows us to express a function f with another function T if f is measurable with respect to T...
(measure theory) - Farkas's lemmaFarkas's lemmaFarkas' lemma is a result in mathematics stating that a vector is either in a given convex cone or that there exists a plane separating the vector from the cone, but not both. It was originally proved by the Hungarian mathematician Gyula Farkas...
(linear programmingLinear programmingLinear programming is a mathematical method for determining a way to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships...
) - Fatou's lemmaFatou's lemmaIn 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...
(measure theory) - Feinstein's fundamental lemma (probability theoryProbability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
) - Fekete's lemmaSubadditive functionIn mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. There are numerous examples of subadditive...
(mathematical analysisMathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
) - Feld–Tai lemmaReciprocity (electromagnetism)In classical electromagnetism, reciprocity refers to a variety of related theorems involving the interchange of time-harmonic electric current densities and the resulting electromagnetic fields in Maxwell's equations for time-invariant linear media under certain constraints...
(electromagnetismElectromagnetismElectromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
) - Finsler's lemma (control theoryControl theoryControl theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
) - Fitting lemmaFitting lemmaThe Fitting lemma, named after the mathematician Hans Fitting, is a basic statement in abstract algebra. Suppose M is a module over some ring...
(abstract algebraAbstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
) - Five lemmaFive lemmaIn mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams....
(homological algebraHomological algebraHomological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
) - Fixed-point lemma for normal functionsFixed-point lemma for normal functionsThe fixed-point lemma for normal functions is a basic result in axiomatic set theory stating that any normal function has arbitrarily large fixed points...
(axiomatic set theory) - Fodor's lemmaFodor's lemmaIn mathematics, particularly in set theory, Fodor's lemma states the following:If \kappa is a regular, uncountable cardinal, S is a stationary subset of \kappa, and f:S\rightarrow\kappa is regressive In mathematics, particularly in set theory, Fodor's lemma states the following:If \kappa is a...
(set theorySet theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
) - Forking lemmaForking lemmaThe forking lemma is any of a number of related lemmas in cryptography research. The lemma states that if an adversary , on inputs drawn from some distribution, produces an output that has some property with non-negligible probability, then with non-negligible probability, if the adversary is...
(cryptographyCryptographyCryptography is the practice and study of techniques for secure communication in the presence of third parties...
) - Frattini's lemmaFrattini's argumentIn group theory, a branch of mathematics, Frattini's argument is an important lemma in the structure theory of finite groups. It is named after Giovanni Frattini, who first used it in a paper from 1885 when defining the Frattini subgroup of a group....
(finite groups) - Friedrichs' lemma
- Frostman's lemma (geometric measure theoryGeometric measure theoryIn mathematics, geometric measure theory is the study of the geometric properties of the measures of sets , including such things as arc lengths and areas. It uses measure theory to generalize differential geometry to surfaces with mild singularities called rectifiable sets...
) - Fundamental lemma (Langlands program)
- Fundamental lemma of calculus of variationsFundamental lemma of calculus of variationsIn mathematics, specifically in the calculus of variations, the fundamental lemma in the calculus of variations is a lemma that is typically used to transform a problem from its weak formulation into its strong formulation .-Statement:A function is said to be of class C^k if it is k-times...
- Fundamental lemma of interpolation theory (numerical analysisNumerical analysisNumerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
) - Fundamental lemma of sieve theoryFundamental lemma of sieve theoryIn number theory, the fundamental lemma of sieve theory is any of several results that systematize the process of applying sieve methods to particular problems...
(sieve theorySieve theorySieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The primordial example of a sifted set is the set of prime numbers up to some prescribed limit X. Correspondingly, the primordial example of a...
) - Gauss's lemmaGauss's lemmaGauss's lemma can mean any of several lemmas named after Carl Friedrich Gauss:* Gauss's lemma * Gauss's lemma * Gauss's lemma...
s (polynomialPolynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s | number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
| Riemannian geometryRiemannian geometryRiemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
) - Glivenko–Cantelli lemma (statisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
) - Gödel's diagonal lemmaDiagonal lemmaIn mathematical logic, the diagonal lemma or fixed point theorem establishes the existence of self-referential sentences in certain formal theories of the natural numbers -- specifically those theories that are strong enough to represent all computable functions...
(mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
) - Goursat's lemmaGoursat's lemmaGoursat's lemma is an algebraic theorem about subgroups of the direct product of two groups.It can be stated as follows.An immediate consequence of this is that the subdirect product of two groups can be described as a fiber product and vice versa....
(algebraAlgebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
) - Gronwall's inequalityGrönwall's inequalityIn mathematics, Gronwall's lemma or Grönwall's lemma, also called Gronwall–Bellman inequality, allows one to bound a function that is known to satisfy a certain differential or integral inequality by the solution of the corresponding differential or integral equation. There are two forms of the...
Gronwall's lemma (inequalities) - Gromov's convex integration lemma
- Gross's integration lemma
- Grothendieck lemma (differential formDifferential formIn the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
s) named after Alexander GrothendieckAlexander GrothendieckAlexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory... - Handshaking lemma (graph theoryGraph theoryIn mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
) - Hardy–Littlewood lemma (differentiationDerivativeIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
) - Harmonic series summation lemma
- Haruki's lemma (plane geometryPlane geometryIn mathematics, plane geometry may refer to:*Euclidean plane geometry, the geometry of plane figures,*geometry of a plane,or sometimes:*geometry of a projective plane, most commonly the real projective plane but possibly the complex projective plane, Fano plane or others;*geometry of the hyperbolic...
) - Hartogs' lemma (several complex variablesSeveral complex variablesThe theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...
) - Hayashi's connecting lemma
- Hensel's lemmaHensel's lemmaIn mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a polynomial equation has a simple root modulo a prime number , then this root corresponds to a unique root of the same equation modulo any higher power...
(commutative ringCommutative ringIn ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
s) - Higman's lemmaHigman's lemmaIn mathematics, Higman's lemma states that the set of finite sequences over a finite alphabet, as partially ordered by the subsequence relation, is well-quasi-ordered...
(order theoryOrder theoryOrder theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...
) - Hindley–Rosen lemma
- Horseshoe lemmaHorseshoe lemmaIn homological algebra, the horseshoe lemma, also called the simultaneous resolution theorem, is a statement relating resolutions of two objects A' and A to resolutions ofextensions of A' by A...
(homological algebraHomological algebraHomological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
) - Hotelling's lemmaHotelling's lemmaHotelling's lemma is a result in microeconomics that relates the supply of a good to the profit of the good's producer. It was first shown by Harold Hotelling, and is widely used in the theory of the firm. The lemma is very simple, and can be stated:...
(envelope theoryEnvelope theoremThe envelope theorem is a theorem about optimization problems in microeconomics. It may be used to prove Hotelling's lemma, Shephard's lemma, and Roy's identity...
microeconomics) - Hua's lemmaHua's lemmaIn mathematics, Hua's lemma, named for Hua Loo-keng, is an estimate for exponential sums.It states that if P is an integral-valued polynomial of degree k, \varepsilon is a positive real number, and f a real function defined by...
(analytic number theoryAnalytic number theoryIn mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...
) - Huet's strong confluence lemma
- Injective test lemmaInjective moduleIn mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers...
(homological algebraHomological algebraHomological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
) - Integration lemmas, of which there are several
- Iteration lemmas, of which there are several
- Itō's lemmaIto's lemmaIn 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...
(stochastic calculusStochastic calculusStochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes...
) - Johnson–Lindenstrauss lemmaJohnson–Lindenstrauss lemmaIn mathematics, the Johnson–Lindenstrauss lemma is a result concerning low-distortion embeddings of points from high-dimensional into low-dimensional Euclidean space. The lemma states that a small set of points in a high-dimensional space can be embedded into a space of much lower dimension in such...
(Euclidean geometryEuclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
) - Jordan's lemmaJordan's lemmaIn complex analysis, Jordan's lemma is a result frequently used in conjunction with the residue theorem to evaluate contour integrals and improper integrals...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
)
K to O
- Kalman–Yakubovich–Popov lemma (system analysisSystem analysisSystem analysis in the field of electrical engineering characterizes electrical systems and their properties. System Analysis can be used to represent almost anything from population growth to audio speakers, electrical engineers often use it because of its direct relevance to many areas of their...
, control theoryControl theoryControl theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
) - Kelly's lemma (graph theoryGraph theoryIn mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
) - Klop's lemma (lambda calculusLambda calculusIn mathematical logic and computer science, lambda calculus, also written as λ-calculus, is a formal system for function definition, function application and recursion. The portion of lambda calculus relevant to computation is now called the untyped lambda calculus...
) - Knaster–Kuratowski–Mazurkiewicz lemma (fixed-point theoryFixed-point theoremIn mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point , under some conditions on F that can be stated in general terms...
) - Knuth's 0-1 sorting lemma
- König's lemmaKönig's lemmaKönig's lemma or König's infinity lemma is a theorem in graph theory due to Dénes Kőnig . It gives a sufficient condition for an infinite graph to have an infinitely long path. The computability aspects of this theorem have been thoroughly investigated by researchers in mathematical logic,...
(graph theoryGraph theoryIn mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
} - Kronecker's lemmaKronecker's lemmaIn mathematics, Kronecker's lemma is a result about the relationship between convergence of infinite sums and convergence of sequences. The lemma is often used in the proofs of theorems concerning sums of independent random variables such as the strong Law of large numbers...
(infinite sums) - Krull's separation lemma
- Lambda lemma for normally hyperbolic invariant manifoldManifoldIn mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s (topologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
) - Lax–Milgram lemma (differential equationDifferential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s) - Lebesgue's number lemmaLebesgue's number lemmaIn topology, Lebesgue's number lemma, named after Henri Lebesgue, is a useful tool in the study of compact metric spaces. It states:Such a number δ is called a Lebesgue number of this cover. The notion of a Lebesgue number itself is useful in other applications as well.-References:-External links:*...
(dimension theoryDimension theoryIn mathematics, dimension theory is a branch of general topology dealing with dimensional invariants of topological spaces.-See also:*Lebesgue covering dimension*Inductive dimensions *Dimension...
) - Leftover hash-lemmaLeftover hash-lemmaThe leftover hash lemma is a lemma in cryptography first stated by Russell Impagliazzo, Leonid Levin, and Michael Luby.Imagine that you have a secret key X that has n uniform random bits, and you would like to use this secret key to encrypt a message. Unfortunately, you were a bit careless with the...
(cryptographyCryptographyCryptography is the practice and study of techniques for secure communication in the presence of third parties...
) - Lindelöf's lemmaLindelöf's lemmaIn mathematics, Lindelöf's lemma is a simple but useful lemma in topology on the real line, named for the Finnish mathematician Ernst Leonard Lindelöf.-Statement of the lemma:Let the real line have its standard topology...
(topologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
) - Lindenbaum's lemmaLindenbaum's lemmaIn mathematical logic, Lindenbaum's lemma states that any consistent theory of predicate logic can be extended to a complete consistent theory. It is used in the proof of Gödel's completeness theorem, among other places...
(mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
) - Lions' lemma
- Little's lemma (queuing theory)
- Littlewood–Offord lemma (combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Łojasiewicz factorization lemma
- Lovász local lemmaLovász local lemmaIn probability theory, if a large number of events are all independent of one another and each has probability less than 1, then there is a positive probability that none of the events will occur...
(probability theoryProbability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
) - Margulis lemmaMargulis lemmaIn mathematics, the Margulis lemma is a result about discrete subgroups of isometries of a symmetric space , or more generally a space of non-positive curvature....
(hyperbolic geometryHyperbolic geometryIn mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
) - Matrix determinant lemmaMatrix determinant lemmaIn mathematics, in particular linear algebra, the matrix determinant lemma computes the determinant of the sum of an invertiblematrix Aand the dyadic product, u vT,of a column vector u and a row vector vT.- Statement :...
(matrix theory) - Matrix inversion lemmaWoodbury matrix identityIn mathematics , the Woodbury matrix identity, named after Max A. Woodbury says that the inverse of a rank-k correction of some matrix can be computed by doing a rank-k correction to the inverse of the original matrix...
- Mautner's lemmaMautner's lemmaIn mathematics, Mautner's lemma in representation theory states that if G is a topological group and π a unitary representation of G on a Hilbert space H, then for any x in G, which has conjugates...
(representation theoryRepresentation theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
) - Morse lemmaMorse theoryIn differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a differentiable function on a manifold will, in a typical case, reflect...
(differential topologyDifferential topologyIn mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
) - Moschovakis' coding lemma (set theorySet theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
) - Mostowski collapse lemma (mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
) - Nakayama lemmaNakayama lemmaIn mathematics, more specifically modern algebra and commutative algebra, Nakayama's lemma also known as the Krull–Azumaya theorem governs the interaction between the Jacobson radical of a ring and its finitely generated modules...
(commutative algebraCommutative algebraCommutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
) - Newman's lemmaNewman's lemmaIn the theory of rewriting systems, Newman's lemma states that a terminating abstract rewriting system , that is, one in which there are no infinite reduction sequences, is confluent if it is locally confluent...
(term rewritingRewritingIn mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a formula with other terms. What is considered are rewriting systems...
) - Neyman–Pearson lemma (statisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
) - Nine lemmaNine lemmaIn mathematics, the nine lemma is a statement about commutative diagrams and exact sequences valid in any abelian category, as well as in the category of groups. It states: ifis a commutative diagram and all columns as well as the two bottom rows are exact, then the top row is exact as well...
(homological algebraHomological algebraHomological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
) - Noether's normalization lemma (commutative algebraCommutative algebraCommutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
) - Ogden's lemmaOgden's lemmaIn the theory of formal languages, Ogden's lemma provides an extension of flexibility over the pumping lemma for context-free languages....
(formal languages)
P to T
- Parallel moves lemma
- Parity lemmas, of which there are several
- Partition lemmas, of which there are several
- Ping-pong lemmaPing-pong lemmaIn mathematics, the ping-pong lemma, or table-tennis lemma, is any of several mathematical statements that ensure that several elements in a group acting on a set freely generate a free subgroup of that group.-History:...
(geometric group theoryGeometric group theoryGeometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
) - Piling-up lemmaPiling-up lemmaIn cryptanalysis, the piling-up lemma is a principle used in linear cryptanalysis to construct linear approximation to the action of block ciphers...
(linear cryptanalysisLinear cryptanalysisIn cryptography, linear cryptanalysis is a general form of cryptanalysis based on finding affine approximations to the action of a cipher. Attacks have been developed for block ciphers and stream ciphers...
) - Poincaré lemma of closed and exact differential formsClosed and exact differential formsIn mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero , and an exact form is a differential form that is the exterior derivative of another differential form β...
(differential formDifferential formIn the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
s) - Pólya–Burnside lemma
- Pugh's closing lemmaPugh's closing lemmaIn mathematics, Pugh's closing lemma is a result that links periodic orbit solutions of differential equations to chaotic behaviour. It can be formally stated as follows:-Interpretation:...
- Pumping lemmaPumping lemmaIn the theory of formal languages in computability theory, a pumping lemma or pumping argument states that, for a particular language to be a member of a language class, any sufficiently long string in the language contains a section, or sections, that can be removed, or repeated any number of...
(formal languageFormal languageA formal language is a set of words—that is, finite strings of letters, symbols, or tokens that are defined in the language. The set from which these letters are taken is the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar...
s) sometimes called the Bar-HillelYehoshua Bar-HillelYehoshua Bar-Hillel was an Israeli philosopher, mathematician, and linguist at the Hebrew University of Jerusalem, best known for his pioneering work in machine translation and formal linguistics.- Biography :...
lemma - Quantifier reversal lemma
- Racah factorization lemma
- Rasiowa–Sikorski lemmaRasiowa–Sikorski lemmaIn axiomatic set theory, the Rasiowa–Sikorski lemma is one of the most fundamental facts used in the technique of forcing. In the area of forcing, a subset D of a forcing notion is called dense in P if for any p ∈ P there is d ∈ D with d ≤ p...
(set theorySet theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
) - Recursion lemmas, of which there are several
- Reduction lemmas, of which there are several
- Ricci's lemma (tensorTensorTensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
s) - Riemann–Lebesgue lemma (harmonic analysisHarmonic analysisHarmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
) - Rigidity lemma (algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
) - Riesz's lemmaRiesz's lemmaRiesz'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 :...
(functional analysisFunctional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
) - Rouche–Kronecker–Campelli lemma (linear algebraLinear algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
) - Sard's lemmaSard's lemmaSard's theorem, also known as Sard's lemma or the Morse–Sard theorem, is a result in mathematical analysis which asserts that the image of the set of critical points of a smooth function f from one Euclidean space or manifold to another has Lebesgue measure 0 – they form a null set...
(mathematical analysisMathematical analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, singularity theorySingularity theory-The notion of singularity:In mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping it on the floor...
) - Satisfiability coding lemma
- Schanuel's lemmaSchanuel's lemmaIn mathematics, especially in the area of algebra known as module theory, Schanuel's lemma, namead after Stephen Schanuel, allows one to compare how far modules depart from being projective...
(projective moduleProjective moduleIn mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...
s) - Schreier's subgroup lemmaSchreier's subgroup lemmaSchreier's subgroup lemma is a theorem in group theory used in the Schreier–Sims algorithm and also for finding a presentation of a subgroup.-Definition:Suppose H is a subgroup of G, which is finitely generated with generating set S, that is, G = ....
(group theoryGroup theoryIn mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
) - Schur's lemmaSchur's lemmaIn mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if M and N are two finite-dimensional irreducible representations...
(representation theoryRepresentation theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
) - Schwarz lemmaSchwarz lemmaIn mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than stronger theorems, such as the Riemann mapping theorem, which it helps to prove...
(complex analysisComplex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
) - Schwartz–Zippel lemma (polynomialPolynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s) - Separation lemmas, of which there are several
- Shadowing lemmaShadowing lemmaIn the theory of dynamical systems, the shadowing lemma is a lemma describing the behaviour of pseudo-orbits near a hyperbolic invariant set. Informally, it says that every pseudo-orbit stays uniformly close to some true trajectory - in other words, a...
(geometryGeometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
) - Shephard's lemmaShephard's lemmaShephard's lemma is a major result in microeconomics having applications in the theory of the firm and in consumer choice. The lemma states that if indifference curves of the expenditure or cost function are convex, then the cost minimizing point of a given good with price p_i is unique...
(microeconomicsMicroeconomicsMicroeconomics is a branch of economics that studies the behavior of how the individual modern household and firms make decisions to allocate limited resources. Typically, it applies to markets where goods or services are being bought and sold...
) - Short five lemmaShort five lemmaIn mathematics, especially homological algebra and other applications of abelian category theory, the short five lemma is a special case of the five lemma....
(homological algebraHomological algebraHomological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
) - Siegel's lemmaSiegel's lemmaIn transcendental number theory and Diophantine approximation, Siegel's lemma refers to bounds on the solutions of linear equations obtained by the construction of auxiliary functions. The existence of these polynomials was proven by Axel Thue; Thue's proof used Dirichlet's box principle. Carl...
(Diophantine approximationDiophantine approximationIn number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers....
) - Snake lemmaSnake lemmaThe snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance in algebraic topology...
(homological algebraHomological algebraHomological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
) - Sperner's lemmaSperner's lemmaIn mathematics, Sperner's lemma is a combinatorial analog of the Brouwer fixed point theorem, which follows from it. Sperner's lemma states that every Sperner coloring of a triangulation of an n-dimensional simplex contains a cell colored with a complete set of colors...
(combinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
) - Splitting lemmaSplitting lemmaIn mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements for short exact sequence are equivalent....
(homological algebraHomological algebraHomological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
) - Stein's lemmaStein's lemmaStein's lemma, named in honor of Charles Stein, is a theorem of probability theory that is of interest primarily because of its applications to statistical inference — in particular, to James–Stein estimation and empirical Bayes methods — and its applications to portfolio choice...
(probability theoryProbability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
) - Stewart–Walker lemma (tensorTensorTensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
s) - Szemerédi regularity lemmaSzemerédi regularity lemmaIn mathematics, the Szemerédi regularity lemma states that every large enough graph can be divided into subsets of about the same size so that the edges between different subsets behave almost randomly. introduced a weaker version of this lemma, restricted to bipartite graphs, in order to prove ...
(graph theoryGraph theoryIn mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
) - Transformation lemmas, of which there are several
- Tube lemmaTube lemmaIn mathematics, particularly topology, the tube lemma is a useful tool in order to prove that the finite product of compact spaces is compact. It is in general, a concept of point-set topology.-Tube lemma:...
(topologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
) - Tukey's lemmaTukey's lemmaIn mathematics, Tukey's lemma, named after John Tukey, states that every nonempty collection of finite character has a maximal element with respect to inclusion. It is equivalent to the Axiom of Choice.- References :* Brillinger, David R. "John Wilder Tukey"...
(metamathematicsMetamathematicsMetamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories...
) also known as the Teichmüller-Tukey lemma
U to Z
- Ultrafilter lemma (order theoryOrder theoryOrder theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...
) - Uniform bounding lemma
- Urysohn's lemmaUrysohn's lemmaIn 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....
(general topologyGeneral topologyIn mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...
) - Varadhan's integration lemma
- Vaughan's lemma (analytic number theoryAnalytic number theoryIn mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...
) - Verdu–Han lemma (probability theoryProbability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
) - Vitali covering lemmaVitali covering lemmaIn mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces. This lemma is an intermediate step, of independent interest, in the proof of the Vitali covering theorem. The covering theorem is credited to the Italian...
(real analysisReal analysisReal analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
) - Vizing's adjacency lemma (graph theoryGraph theoryIn mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
) - Wald's lemma (probability theoryProbability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
) - Watson's lemmaWatson's lemmaIn mathematics, Watson's lemma, proved by G. N. Watson , has significant application within the theory on the asymptotic behavior of integrals.- Statement of the lemma :...
- Weyl's lemma (Laplace equation)Weyl's lemma (Laplace equation)In mathematics, Weyl's lemma is a result that provides a "very weak" form of the Laplace equation. It is named after the German mathematician Hermann Weyl.-Statement of the lemma:...
(partial differential equations) - Whitehead's lemma (Lie algebraLie algebraIn mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
s) - Yao's XOR lemma (cryptographyCryptographyCryptography is the practice and study of techniques for secure communication in the presence of third parties...
) - Yoneda lemmaYoneda lemmaIn mathematics, specifically in category theory, the Yoneda lemma is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory...
, (category theoryCategory theoryCategory theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
) - Zassenhaus lemmaZassenhaus lemmaIn mathematics, the butterfly lemma or Zassenhaus lemma, named after Hans Julius Zassenhaus, is a technical result on the lattice of subgroups of a group or the lattice of submodules of a module, or more generally for any modular lattice....
(group theoryGroup theoryIn mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
) - Zolotarev's lemmaZolotarev's lemmaIn number theory, Zolotarev's lemma states that the Legendre symbol\leftfor an integer a modulo an odd prime number p, where p does not divide a, can be computed as the sign of a permutation:...
(number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) - Zorn's lemmaZorn's lemmaZorn'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...
also known as the Kuratowski–Zorn lemma (set theorySet theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
)