# Category: Lemmas

Trilemma
A trilemma is a difficult choice from three options, each of which is (or appears) unacceptable or unfavourable. There are two logically equivalent ways in which to express a trilemma: it can be expre
Tucker's lemma
In mathematics, Tucker's lemma is a combinatorial analog of the Borsuk–Ulam theorem, named after Albert W. Tucker. Let T be a triangulation of the closed n-dimensional ball . Assume T is antipodally s
Tube lemma
In mathematics, particularly topology, the tube lemma is a useful tool in order to prove that the finite product of compact spaces is compact.
Cartan's lemma
No description available.
Lebesgue's number lemma
In topology, Lebesgue's number lemma, named after Henri Lebesgue, is a useful tool in the study of compact metric spaces. It states: If the metric space is compact and an open cover of is given, then
Ring lemma
In the geometry of circle packings in the Euclidean plane, the ring lemma gives a lower bound on the sizes of adjacent circles in a circle packing.
Dickson's lemma
In mathematics, Dickson's lemma states that every set of -tuples of natural numbers has finitely many minimal elements. This simple fact from combinatorics has become attributed to the American algebr
Dynkin's lemma
No description available.
Liberman's lemma
Liberman's lemma is a theorem used in studying intrinsic geometry of convex surface.It is named after Joseph Liberman.
In mathematics, Varadhan's lemma is a result from large deviations theory named after S. R. Srinivasa Varadhan. The result gives information on the asymptotic distribution of a statistic φ(Zε) of a fa
Book of Lemmas
The 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 (lemmas) on circles.
Margulis lemma
In differential geometry, the Margulis lemma (named after Grigory Margulis) is a result about discrete subgroups of isometries of a non-positively curved Riemannian manifold (e.g. the hyperbolic n-spa
Ky Fan lemma
In mathematics, Ky Fan's lemma (KFL) is a combinatorial lemma about labellings of triangulations. It is a generalization of Tucker's lemma. It was proved by Ky Fan in 1952.
Schur's lemma
In 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 irre
Open and closed maps
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function is open if for any open set in the image i
Kronecker's lemma
In mathematics, Kronecker's lemma (see, e.g., , Lemma IV.3.2)) is a result about the relationship between convergence of infinite sums and convergence of sequences. The lemma is often used in the proo
Perfect group
In mathematics, more specifically in group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients (equivalen
Quillen's lemma
In algebra, Quillen's lemma states that an endomorphism of a simple module over the enveloping algebra of a finite-dimensional Lie algebra over a field k is algebraic over k. In contrast to a version
F. Riesz's theorem
F. Riesz's theorem (named after Frigyes Riesz) is an important theorem in functional analysis that states that a Hausdorff topological vector space (TVS) is finite-dimensional if and only if it is loc
Pumping lemma for regular languages
In the theory of formal languages, the pumping lemma for regular languages is a lemma that describes an essential property of all regular languages. Informally, it says that all sufficiently long stri
Dwork's lemma
No description available.
Gauss's lemma
Gauss's lemma can mean any of several lemmas named after Carl Friedrich Gauss: * Gauss's lemma (polynomials) – The greatest common divisor of the coefficients is a multiplicative function * Gauss's
Littlewood–Offord problem
In mathematical field of combinatorial geometry, the Littlewood–Offord problem is the problem of determining the number of of a set of vectors that fall in a given convex set. More formally, if V is a
Sauer–Shelah lemma
In combinatorial mathematics and extremal set theory, the Sauer–Shelah lemma states that every family of sets with small VC dimension consists of a small number of sets. It is named after and Saharon
Knaster–Kuratowski–Mazurkiewicz lemma
The Knaster–Kuratowski–Mazurkiewicz lemma is a basic result in mathematical fixed-point theory published in 1929 by Knaster, Kuratowski and Mazurkiewicz. The KKM lemma can be proved from Sperner's lem
Covering lemma
In the foundations of mathematics, a covering lemma is used to prove that the non-existence of certain large cardinals leads to the existence of a canonical inner model, called the core model, that is
Pumping lemma for context-free languages
In computer science, in particular in formal language theory, the pumping lemma for context-free languages, also known as the Bar-Hillel lemma, is a lemma that gives a property shared by all context-f
Sperner's lemma
In mathematics, Sperner's lemma is a combinatorial result on colorings of triangulations, analogous to the Brouwer fixed point theorem, which is equivalent to it. It states that every Sperner coloring
Craig interpolation
In 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, an
Little's lemma
No description available.
Levi's lemma
In theoretical computer science and mathematics, especially in the area of combinatorics on words, the Levi lemma states that, for all strings u, v, x and y, if uv = xy, then there exists a string w s
Finsler's lemma
Finsler's lemma is a mathematical result named after Paul Finsler. It states equivalent ways to express the positive definiteness of a quadratic form Q constrained by a linear form L. Since it is equi
Hotelling's lemma
Hotelling's lemma is a result in microeconomics that relates the supply of a good to the maximum profit of the producer. It was first shown by Harold Hotelling, and is widely used in the theory of the
In the theory of dynamical systems, the shadowing lemma is a lemma describing the behaviour of pseudo-orbits near a hyperbolic invariant set. Informally, the theory states that every pseudo-orbit (whi
Lax-Milgram lemma
No description available.
Hilbert's lemma
Hilbert's lemma was proposed at the end of the 19th century by mathematician David Hilbert. The lemma describes a property of the principal curvatures of surfaces. It may be used to prove Liebmann's t
Thom's first isotopy lemma
In mathematics, especially in differential topology, Thom's first isotopy lemma states: given a smooth map between smooth manifolds and a closed Whitney stratified subset, if is proper and is a submer
Dehn's lemma
In mathematics, Dehn's lemma asserts that a piecewise-linear map of a disk into a 3-manifold, with the map's singularity set in the disk's interior, implies the existence of another piecewise-linear m
Siegel's lemma
In mathematics, specifically in transcendental number theory and Diophantine approximation, Siegel's lemma refers to bounds on the solutions of linear equations obtained by the construction of auxilia
Algorithmic Lovász local lemma
In theoretical computer science, the algorithmic Lovász local lemma gives an algorithmic way of constructing objects that obey a system of constraints with limited dependence. Given a finite set of ba
List of lemmas
This following is a list of lemmas (or, "lemmata", i.e. minor theorems, or sometimes intermediate technical results factored out of proofs). See also list of axioms, list of theorems and list of conje
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 continuous function. Urysohn's lemma is comm
Danielson-Lanczos lemma
No description available.
Hua's lemma
In 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, is a positive real number, and f a real func
Gauss's lemma (Riemannian geometry)
In Riemannian geometry, Gauss's lemma asserts that any sufficiently small sphere centered at a point in a Riemannian manifold is perpendicular to every geodesic through the point. More formally, let M
Kalman–Yakubovich–Popov lemma
The Kalman–Yakubovich–Popov lemma is a result in system analysis and control theory which states: Given a number , two n-vectors B, C and an n x n Hurwitz matrix A, if the pair is completely controlla
Feld-Tai lemma
No description available.
Dynkin system
A Dynkin system, named after Eugene Dynkin is a collection of subsets of another universal set satisfying a set of axioms weaker than those of 𝜎-algebra. Dynkin systems are sometimes referred to as 𝜆-
Neyman–Pearson lemma
In statistics, the Neyman–Pearson lemma was introduced by Jerzy Neyman and Egon Pearson in a paper in 1933. The Neyman-Pearson lemma is part of the Neyman-Pearson theory of statistical testing, which
Lindenbaum's lemma
In mathematical logic, Lindenbaum's lemma, named after Adolf Lindenbaum, states that any consistent theory of predicate logic can be extended to a complete consistent theory. The lemma is a special ca
Big-little-big lemma
In the mathematics of paper folding, the big-little-big lemma is a necessary condition for a crease pattern with specified mountain folds and valley folds to be able to be folded flat. It differs from
Mackey–Arens theorem
The Mackey–Arens theorem is an important theorem in functional analysis that characterizes those locally convex vector topologies that have some given space of linear functionals as their continuous d
Lindelöf's lemma
In mathematics, Lindelöf's lemma is a simple but useful lemma in topology on the real line, named for the Finnish mathematician Ernst Leonard Lindelöf.
Lindström–Gessel–Viennot lemma
In Mathematics, the Lindström–Gessel–Viennot lemma provides a way to count the number of tuples of non-intersecting lattice paths, or, more generally, paths on a directed graph. It was proved by Gesse
Kac's lemma
In ergodic theory, Kac's lemma, demonstrated by mathematician Mark Kac in 1947, is a lemma stating that in a measure space the orbit of almost all the points contained in a set of such space, whose me
Newman's lemma
In mathematics, in the theory of rewriting systems, Newman's lemma, also commonly called the diamond lemma, states that a terminating (or strongly normalizing) abstract rewriting system (ARS), that is
Borel–Cantelli lemma
In probability theory, the Borel–Cantelli lemma is a theorem about sequences of events. In general, it is a result in measure theory. It is named after Émile Borel and Francesco Paolo Cantelli, who ga
Johnson–Lindenstrauss lemma
In mathematics, the Johnson–Lindenstrauss lemma is a result named after William B. Johnson and Joram Lindenstrauss concerning low-distortion embeddings of points from high-dimensional into low-dimensi
Thom's second isotopy lemma
In mathematics, especially in differential topology, Thom's second isotopy lemma is a family version of Thom's first isotopy lemma; i.e., it states a family of maps between Whitney stratified spaces i
Mostowski collapse lemma
In mathematical logic, the Mostowski collapse lemma, also known as the Shepherdson–Mostowski collapse, is a theorem of set theory introduced by Andrzej Mostowski and.
Hautus lemma
In control theory and in particular when studying the properties of a linear time-invariant system in state space form, the Hautus lemma (after Malo L. J. Hautus), also commonly known as the Popov-Bel
Schur's lemma (Riemannian geometry)
In Riemannian geometry, Schur's lemma is a result that says, heuristically, whenever certain curvatures are pointwise constant then they are forced to be globally constant. The proof is essentially a
Gordan's lemma
Gordan's lemma is a lemma in convex geometry and algebraic geometry. It can be stated in several ways. * Let be a matrix of integers. Let be the set of non-negative integer solutions of . Then there
Itô's lemma
In mathematics, Itô's lemma or Itô's formula (also called the Itô-Doeblin formula, especially in French literature) is an identity used in Itô calculus to find the differential of a time-dependent fun
Higman's lemma
In 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. That is, if is an infinite seque
Morse theory
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insi
Krull's separation lemma
In abstract algebra, Krull's separation lemma is a lemma in ring theory. It was proved by Wolfgang Krull in 1928.
Lovász local lemma
In 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 (possibly small) probability that none of the event
Piling-up lemma
In cryptanalysis, the piling-up lemma is a principle used in linear cryptanalysis to construct linear approximations to the action of block ciphers. It was introduced by Mitsuru Matsui (1993) as an an
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,
Ogden's lemma
In the theory of formal languages, Ogden's lemma (named after ) is a generalization of the pumping lemma for context-free languages.
Diagonal lemma
In mathematical logic, the diagonal lemma (also known as diagonalization lemma, self-reference lemma or fixed point theorem) establishes the existence of self-referential sentences in certain formal t
Pyrrho's lemma
In statistics, Pyrrho's lemma is the result that if one adds just one extra variable as a regressor from a suitable set to a linear regression model, one can get any desired outcome in terms of the co
Isolation lemma
In theoretical computer science, the term isolation lemma (or isolating lemma) refers to randomized algorithms that reduce the number of solutions to a problem to one, should a solution exist.This is
Tetralemma
The tetralemma is a figure that features prominently in the logic of India.
Counting lemma
The counting lemmas this article discusses are statements in combinatorics and graph theory. The first one extracts information from -regular pairs of subsets of vertices in a graph , in order to guar
Slepian's lemma
In probability theory, Slepian's lemma (1962), named after David Slepian, is a Gaussian comparison inequality. It states that for Gaussian random variables and in satisfying , the following inequality
Interchange lemma
In the theory of formal languages, the interchange lemma states a necessary condition for a language to be context-free, just like the pumping lemma for context-free languages. It states that for ever
Robbins lemma
In statistics, the Robbins lemma, named after Herbert Robbins, states that if X is a random variable having a Poisson distribution with parameter λ, and f is any function for which the expected value