List of math theorems

Important mathematical theorems usually have a distinctive name by which they are often internationally known. This list gives a brief indication of the content of the sentence for each such name, further details can then be found in the respective articles. The alphabetical sorting of sets listed below always based on the name of the mathematician, if one is connected to the set that Whitney embedding theorem can be found therefore under W and not E . Terms such as "set" or "Lemma" are never sort criteria (but "Fundamental Theorem" or "law"), the closed graph theorem can be found therefore under A .

Many inequalities also enjoy the rank of a mathematical proposition, only the most important are represented here, others can be found in the category of inequalities .

0-9

• 123 Theorem : An estimate of the difference between independent, identically distributed random variables

A.

• Set of Abel-Ruffini : a general polynomial of degree five or greater is resolvable not by radicals.
• Abel shear limit theorem : Theorem on the convergence of a power series in the border point of convergence interval.
• Abel 's lemma : Absolute and locally uniform convergence of power series
• Abel -specific partial summation : forming the sum of products
• Theorem of the closed picture : Conditions under which the picture of a continuous, linear operator between Banach spaces is closed
• Theorem of the closed graph : A linear operator with a closed graph between Banach spaces is continuous
• Ado's theorem : Representability of Lie algebras as matrices
• Theorem of Alexander : Every loop is the end of a braid.
• Alexander theorem : It suffices to check compactness with the sets of a sub-basis.
• Tori set of Alexander : Every differentiable in S ³ embedded torus bounded one in S ³ embedded Volltorus.
• Decomposition theorem of Alexandroff-Borsuk : Characterization of decomposing compacts in Euclidean space
• Theorem of Alexandroff-Urysohn : Every compact Polish space is a constant image of the Cantor space.
• Andersen-Jessen theorem : Existence of measures in uncountable product spaces.
• Anne's theorem : About the area division of a convex quadrilateral, which is not a parallelogram, by a point on the Newton line.
• Anosov's closure lemma : closed pseudo-orbites can be approximated by periodic orbites.
• Embedding theorem by Arens-Eells : embedding metric spaces in standardized vector spaces
• Set of Artin - Rees : sentence about powers of ideals and finitely generated modules of a Noetherian ring.
• Artin - Wedderburn's theorem : Every semi-simple ring is isomorphic to a direct product of matrix rings over oblique bodies.
• Artin 's law of reciprocity : The Galois group of an Abelian body extension is the quotient of an ideal class group
• Set of Arzelà-Ascoli : compactness of continuous functions in rooms
• Atiyah - Bott Fixed Point Theorem : Calculation of the Lefschetz number for endomorphisms of elliptical complexes
• Set of Atiyah - Jänich The K: ₀ is a space group is isomorphic to the group of homotopy classes of functions with values in the Fredholm operators.
• Atiyah - Singer index theorem : equality of analytic and topological index for elliptic differential operators on compact manifolds
• Atkinson's theorem : An operator is Fredholm operator if and only if there is an operator such that and are compact. ${\displaystyle T}$ ${\displaystyle S}$ ${\displaystyle 1-TS}$ ${\displaystyle 1-ST}$

• Apollonios' theorem : Relationship between squares at a triangle
• Lemma von Auerbach : Existence of an Auerbach basis in finite-dimensional spaces
• Theorem of the excluded third party : an axiom or axiom scheme in logic
• Exterior angle theorem : exterior angles in Euclidean triangles

B.

• Babai's theorem : a theorem about the class of all finite, simple graphs
• Baer - Epstein theorem : homotopic curves on surfaces are isotopic, homotopic homeomorphisms of surfaces are isotopic.
• Baire's theorem (category theorem ): Countable averages of open, dense sets in complete spaces are dense.
• Balinski and Young's Impossibility Theorem: A phrase about seat allocation procedures
• Banach 's mapping theorem : For functionsandthere are disjoint decompositions,withand ${\displaystyle f\colon M\to N}$ ${\displaystyle g\colon N\to M}$ ${\displaystyle M\!=\!M_{1}\!\cup \!M_{2}}$ ${\displaystyle N\!=\!N_{1}\!\cup \!N_{2}}$ ${\displaystyle f(M_{1})\!=\!N_{1}}$ ${\displaystyle g(N_{2})\!=\!M_{2}}$
• Set of Banach - Alaoglu : weak - * - compactness of the unit ball in dual space
• Banach's Fixed Point Theorem : Every contracting mapping on a non-empty, complete, metric space has exactly one fixed point.
• Banach's Theorem - Dieudonné : A subspace in the dual space of a Banach space is weakly - * - closed if and only if its unit sphere is it.
• Banach - Mackey theorem : Every weakly-bounded Banach sphere in a locally convex space is strongly-bounded.
• Banach - Mazur theorem : Every separable Banach space is isometrically isomorphic to a subspace of . ${\displaystyle C([0,1])}$
• Banach's Theorem - Steinhaus : Principle of uniform limitation
• Banach - Stone's theorem : Characterization of compact Hausdorff spaces through their continuous functions.
• Barankin and Stein Theorem : Characterization of the locally minimal unbiased estimators.
• Set of Baranyai : The full Hyper graph on node whose hyperedges always nodes connect, has a 1-factorization if and only if a divisor of is. ${\displaystyle n}$ ${\displaystyle m}$ ${\displaystyle m}$ ${\displaystyle n}$
• Theorems of Basu : Theorems about relationships between sufficiency, completeness and freedom of distribution in statistics
• Basic selection theorem : Every generating system of a vector space contains a basis.
• Bauer-Fike theorem (numerical mathematics): Provides an estimate of the change in the eigenvalues ​​of matrices with respect to perturbations
• Set of Bayes : enables the calculation of the conditional probability of . ${\displaystyle P(A|B)}$ ${\displaystyle P(B|A)}$
• Beckman and Quarles theorem : Geometric transformations in -dimensional spaces, characterization of isometries ${\displaystyle n}$
• Beker's theorem in finite geometry: The strongly resolvable 3-block plans are exactly the Hadamard 3-block plans.
• Set of Beltrami - Enneper : Correlation between torsion and Gaussian curvature of a surface extending in a curve.
• Bernoulli's Theorem : Several sentences going back to members of the Bernoulli family
• Bernstein inequality (stochastics): Upper bound for the probability that the arithmetic mean of random variables exceeds a given value
• Bernstein Inequalities (Analysis): Upper bounds for the derivation of polynomials in a closed interval
• Bernstein - von-Mises theorem : statement of Bayesian statistics
• Set of BlackBerry Esseen : Set on the quality of convergence in the central limit theorem
• Bertrand 's postulate : For every natural numberthere is a prime numberwith. ${\displaystyle n}$ ${\displaystyle p}$ ${\displaystyle n<p\leq 2n}$
• Bessaga - Pelczynski selection principle : For the selection of basic sequences from certain sequences in Banach spaces
• Lemma von Bézout : This can be represented as a linear combination of and with integer coefficients. ${\displaystyle \operatorname {ggT} (a,b)}$ ${\displaystyle a}$ ${\displaystyle b}$
• Set of Bézout : Two plane curves of degree or intersect at (counted with multiplicities) points. ${\displaystyle d}$ ${\displaystyle e}$ ${\displaystyle d\cdot e}$
• Bieberbach's Hypothesis : A now proven theorem about coefficient estimates of certain holomorphic functions
• Bienaymé's equation : The variance of a sum of uncorrelated random variables is equal to the sum of their variances.
• Bicommutant Theorem : A Von Neumann algebra agrees with its double commutant.
• Binet - Cauchy theorem : Calculation of the determinant of a square matrix given as a product
• Set of Bing - Nagata - Smirnov : theorem on the topological spaces Metrisierbarkeit
• Bipolar theorem : The bipolar of a set is equal to its absolutely convex, weakly closed envelope.
• Illustration set of Birkhoff : Each algebra is isomorphic to a product subdirect subdirectly irreducible algebras of the same type.
• Theorem of Birkhoff and von Neumann : The permutation matrices are exactly the extreme points of the double-stochastic matrices.
• Bishop - de Leeuw's theorem : To represent points of a compact, convex set by probability measures on the extremal points.
• Blackwell's renewal theorem: A theorem from renewal theory about the asymptotically expected number of renewals in a time interval.
• Blaschke's selection sentence . The space of the non-empty compact convex subsets of a normalized vector space is locally compact with respect to the Hausdorff metric.
• Blaschke's convergence theorem : Sufficient condition for compact convergence of a series of holomorphic functions on the unit circle.
• Bloch's theorem : A theorem about image domains of holomorphic functions
• Blumenthal 's zero-one law : In a Wiener process with filtration, an event from hasthe probability 0 or 1. ${\displaystyle (F_{t})_{t}}$ ${\displaystyle F_{0}^{+}}$
• Bochner's theorem : A continuous function is a characteristic function of a probability measure if and only if it is positive semidefinite with the value 1 at the position 0.
• Set of Bohr - Mollerup : Characterization of the gamma function by means of logarithmic convexity
• Bolyai - Gerwien theorem : Plane polygons of the same area can be broken down into a finite number of congruent triangles.
• Bonse's inequality : the square of a prime number is smaller than the product of all smaller prime numbers
• Set of Bolzano-Weierstrass : Every bounded sequence of real numbers contains at least one convergent subsequence.
• Bonnet - Myers theorem : Every complete, connected Riemannian manifold with "downwardly bounded Ricci tensor" is compact with a finite fundamental group.
• Lemma von Borel - Cantelli : Theorem from probability theory about the Limes Superior of events
• Theorem of Borsuk - Ulam : Theorem about continuous functions on the -sphere (antipodal points) ${\displaystyle n}$
• Bose's theorem : The proposition formulates necessary conditions for the existence of a block plan with parallelism. In this case, the theorem exacerbates Fisher's inequality .
• Set of Brahmagupta : sentence about track conditions in certain tendons squares
• Lemma von Bramble - Hilbert : Estimation of the error in an approximation by polynomials in Sobolew spaces
• Brauer - Suzuki's theorem : A criterion that the center of the group is of order 2.
• Set of Brianchon : set on the diagonal point of intersection of a hexagon circumscribed a conic
• Theorem of the British flag : generalization of the Pythagorean theorem
• Brooks's Theorem : The vertex-coloring number of a connected graph that is neither complete nor a circle of odd length is at most as high as the maximum degree of the graph.
• Brouwer's fixed point theorem : Every continuous mapping of the -dimensional solid sphere into the -dimensional solid sphere has a fixed point. ${\displaystyle n}$ ${\displaystyle n}$
• Bruck - Ryser - Chowla theorem : Necessary condition for the existence of certain block plans.
• Brunn-Minkowski inequality : relationship between the Lebesgue measure of two sets and the Lebesgue measure of their Minkowski sum.
• Büchi theorem : The MSO-definable languages ​​are exactly the regular languages.
• Burnside's theorem : Finite groups of order p ^a q ^b are solvable.
• Bundle theorem : Characterization ovoidaler Mobius levels

C.

• Lemma from Calderón - Zygmund : Decomposition of integrable functions.
• Cantelli's theorem : Sufficient conditions under which the strong law of large numbers holds.
• Set of Cantor : A lot has always a smaller thickness than its power set.
• Set of Cantor - Bendixson Theorem on discharges of sets in topological spaces
• Theorem of Cantor-Bernstein-Schröder : If a set is at most equal to a set and at most equal to then are and equal. ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle A,}$ ${\displaystyle A}$ ${\displaystyle B}$
• Sechsecksatz of Catalan : Characterization of hexagons lying on a circle corners
• Continued set of Carathéodory : Continuation of conformal mappings on the edge simply-connected, open, Jordan bordered deterministic quantities.
• Carathéodory's extension theorem : continuation of measures on quantity rings to measures on σ-algebras
• Carmichael's theorem : Conditions for a multiplicative congruence generator with maximum period length.
• Set of Carnot : set via home and within the radius of a triangle
• Set of Carnot : sentence about solders triangle sides
• Carnot's theorem : relationship between conics and triangles
• Cartan's Theorem : Completed Subgroups of Lie Groups
• Set of Cartan - Ambrose - Hicks : Riemannian metrics are determined locally by Riemannian curvature tensor.
• Cartan - Hadamard theorem : The exponential mapping over a complete Riemannian manifold of non-positive intersectional curvature is a superposition.
• Cartan criterion : Criteria for solvability and semi-simplicity of Lie algebras
• Casey's theorem : Geometric theorem about tangent sections of four circles in a certain configuration
• Catalan conjecture : Meanwhile proven number theoretic statement about powers with difference 1.
• Cauchy's Theorem (Geometry) : Theorem on the areas of projections of convex sets
• Cauchy's theorem (group theory) : A finite group contains an element of this order for every prime divisor of its order ${\displaystyle p}$ ${\displaystyle p}$
• Cauchy 's limit theorem : Theorem about the convergence of the arithmetic mean of a convergent sequence
• Cauchy's integral formula : Basic integral representation of holomorphic functions
• Cauchy 's integral theorem : Theorem about the vanishing of integrals of holomorphic functions over closed paths
• Cauchy 's mean value theorem (also mean value theorem of integral calculus )
• Cauchy product formula : The product of absolutely-convergent series is again absolutely-convergent.
• Cauchy - Hadamard theorem : Formula for determining the radius of convergence of a series
• Cauchy's theorem - Kowalewskaja : Existence and uniqueness theorem for the Cauchy problem with analytical data.
• Cauchy - Schwarz's inequality : The magnitude of the scalar product of two vectors is at most as large as the product of their norms.
• Cayley formula : There arevarious labeled trees withnodes. ${\displaystyle n^{n-2}}$ ${\displaystyle n}$
• Cayley's Theorem : Every group is isomorphic to a group of permutations.
• Cayley - Bacharach's Theorem : A cubic curve that passes through eight of nine points of intersection of two further cubic curves also contains the ninth point; as well as generalizations to higher order curves.
• Cayley-Hamilton theorem : Every square matrix is ​​the root of its characteristic polynomial.
• Cavalieri's principle : Theorem on volumes of bodies
• Lemma of Céa : Error estimation of finite element approximations of elliptic partial differential equations
• Ceva's theorem : Formula about the partial ratios of triangle sides if three corner transversals intersect at one point
• Cheeger's theorem of finiteness : There are finitely many types of diffeomorphism of Riemannian manifolds for given diameter, volume and curvature bounds.
• Cheeger-Gromov's compactness theorem : Convergence of Riemannian manifolds
• Cheeger-Müller theorem : Equality of Reidemeister torsion and analytic torsion
• Chen's theorem : Every sufficiently large even number can be written as the sum of a prime number and a number with at most two prime factors.
• Chernoff's inequality : upper bound for the probability that a sequence of independent Bernoulli experiments deviates from its expected number of successes
• Chinese remainder theorem : Theorem about simultaneous congruences of integers
• Choquet's theorem : To represent points of a metrizable, compact, convex set by probability measures on the extremal points.
• Church - Rosser theorem : In the lambda calculus there is always one term for two equivalent terms to which both can be reduced.
• Chvátal's Theorem : Solution to the geometric problem of museum guards
• Set of Clairaut : theorem on geodesics on surfaces of revolution
• Clarkson's Theorem : -spaces with are uniformly convex. ${\displaystyle L^{p}}$ ${\displaystyle 1<p<\infty }$
• Set of Clifford : set of elementary geometry to a common point of intersection of circles of triangles
• Set of Cochran : set of analysis of variance
• Theorems of Cohen - Seidenberg : Going up and Going down , two sentences about prime ideal chains in ring extensions.
• Commandino's theorem : The four center lines of a tetrahedron intersect at the center of gravity.
• Cook's Theorem : The satisfiability problem of propositional logic is NP-complete.
• Corrádi lemma : Statements about the necessary number of nodes of a hypergraph, if hyper-edges should have a given number of nodes in common.
• Courant-Fischer theorem : Variational characterization of the eigenvalues ​​of a symmetric or Hermitian matrix.
• Craig Interpolation : A Theorem of Logic about the derivability of theorems
• Cramer's rule : Formula for calculating the solution of a uniquely solvable, linear system of equations using determinants.
• Set of Cramér : theorem on the normal distribution of summands.
• Set of Cramér : set on the convergence of identically distributed random variables.
• Cramér-Wold's theorem : a Borel measure on which is uniquely determined by its one-dimensional projections. ${\displaystyle \mathbb {R} ^{n}}$

D.

• Notation set for Boolean algebras : Every Boolean algebra is isomorphic to a set algebra.
• Notation : Decomposition of probability measures into absolutely continuous, singular continuous and discrete components.
• Dedekind's isomorphism theorem : Every two Peano systems are isomorphic.
• Dedekind's modular law : For subgroups of a group , if . ${\displaystyle K(H\cap L)=(KH)\cap L}$ ${\displaystyle K\leq L}$
• Dedekind's theorem of independence : Homomorphisms with different pairs of values ​​in one field are linearly independent.
• Δ-Lemma : Every uncountable family of finite sets contains an uncountable Δ-system (combinatorial set theory).
• Dembowski - Wagner theorem : Criteria under which a symmetrical block plan is a projective space
• Denjoy's theorem : connected property of the real number line
• Desargues ' theorem describes a historical closure theorem which, from today's perspective, characterizes affine and projective planes that can be described as affine and projective planes of a two-dimensional vector space over a sloping body.
• Descartes' four-circle theorem : relationship between four circles that touch
• Sign rule of Descartes : The number of all positive zeros of a real polynomial is equal to the number of sign changes of its coefficient sequence or smaller than this by an even natural number.
• Determinant product theorem : The determinant of the product of two matrices is equal to the product of the determinants.
• Theorem of Diaconescu - Goodman - Myhill : From the axiom of choice follows the theorem of the excluded third party.
• Diagonal theorem : characterization of parallelograms
• Dilworth theorem : On the power of the greatest anticchains and chain decompositions of semi-ordered sets
• Theorem of dimension invariance : Open subsets of and are for not homeomorphic. ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{m}}$ ${\displaystyle n\neq m}$
• Dini's theorem : Every monotonic sequence of continuous, real functions converging pointwise to a continuous function on a compact space converges uniformly.
• Theorem of Dinostratos : To square the circle by means of the quadratrix of Hippias.
• Dirichlet 's approximation theorem : For everyoneandthere isandwith. ${\displaystyle \alpha \in \mathbb {R} }$ ${\displaystyle N\in \mathbb {N} }$ ${\displaystyle q\in \mathbb {N} ,1\leq q\leq N}$ ${\displaystyle p\in \mathbb {Z} }$ ${\displaystyle \left|q\alpha -p\right|\leq 1/(N+1)}$
• Dirichlet 's unit theorem : Description of the structure of the unit group of the whole ring of an algebraic number field
• Dirichlet 's prime number theorem : There are infinitely many prime numbers that are congruent tomodulo(natural numbers in each case, relatively prime to). ${\displaystyle a}$ ${\displaystyle m}$ ${\displaystyle a}$ ${\displaystyle m}$
• Dixmier's approximation theorem : The closed, convex hull of the unitary conjugate of an element of a Von Neumann algebra intersects the center.
• Doob - Dynkin -Lemma : A sentence about the functional relationship between two random variables.
• Doob decomposition : Every adapted, integrable, stochastic process is the sum of a martingale and a predictable process.
• Three-subgroup lemma : From [A, B, C] = [C, B, A] = 1 it follows [B, C, A] = 1 for subgroups A, B, C of a group.
• Three-Squares Theorem : A natural number is the sum of three squares if and only if it is not of the shape . ${\displaystyle 4^{a}(8b+7)}$
• Continuation of Dugundji : Continuation of continuous, locally convex-valued functions in metric spaces.
• Dushnik - Miller theorem : Every partial order is the intersection of linear orders.
• Dvoretzky's theorem : Every Hilbert space is finitely presentable in every infinite-dimensional Banach space.
• Dvoretzky - Rogers theorem : Existence of unconditionally convergent series, which do not absolutely converge, in infinite-dimensional Banach spaces.
• Dynkin 's π-λ theorem : Relationship between the generated Dynkin system and the generated σ-algebra.

E.

• Easton's Theorem : A theorem about values ​​of the continuum function for regular cardinal numbers.
• Eberlein - Šmulian Theorem : For weakly closed subsets of a Banach space, the terms compact and sequential compact coincide.
• Effros - Handelman - Shen's theorem : Every commutative, imperforate, scaled group with Riesz's decomposition property occurs as a group of an AF-C * algebra. ${\displaystyle K_{0}}$
• Set of gentleman : Surjective, real submersions are fiber bundles.
• Eilenberg - Steenrod uniqueness theorem : If a natural transformation of two homology theories is an isomorphism on all spheres, so also on all finite CW complexes.
• Theorem from Eilenberg - Zilber : The singular chain complex of the product of two spaces is homotopy-equivalent to the tensor product of the singular chain complexes of the spaces.
• Inclusion rule : If a sequence lies between two sequences that tend towards the same limit value, then also converges towards this limit value. ${\displaystyle a_{n}}$ ${\displaystyle a_{n}}$
• Constriction theorem : If a function lies between two functions that tend towards the same limit value, then it also converges towards this limit value. ${\displaystyle f}$ ${\displaystyle f}$
• Eisenstein criterion : Criterion for the irreducibility of polynomials
• Elementary replacement : structural theorem for finitely generated modules over a main ideal ring.
• Engel's theorem : Characterization of nilpotent Lie algebras
• Set of Erdős (set theory): Generalization of the decomposition rate of Sierpiński
• Erdős Theorem (Number Theory): Counterexamples to Polignac's Conjecture
• Erdős - Kac's theorem : The number of different prime factors of a randomly drawn number is approximately normally distributed for large ones. ${\displaystyle \{1,\ldots ,N\}}$ ${\displaystyle N\in \mathbb {N} }$
• Set of Erdős - Ko - Rado : The set is an upper limit to the power of a -Schnittfamilie in an on quantity. ${\displaystyle k}$ ${\displaystyle N}$
• Set of Erdős - Rado : A theorem on partition properties of infinite cardinal numbers
• Erdős - Selfridge theorem : A product of consecutive natural numbers is never a real power in natural numbers.
• Theorem of the complementary parallelograms: If a parallelogram is divided into four partial parallelograms by a diagonal point, then two of them are complementary.
• Individual ergodic theorem : The averaged powers of dimensionally conserving transformations of an integrable function almost certainly converge point by point to the conditional expected value.
• ${\displaystyle {\mathcal {L}}^{p}}$- Ergod theorem : The averaged powers of measure-preserving transformations of a p-integrable function converge in the p-th mean to the conditional expected value.
• Ergodic theorem : collective term for several statements of the ergodic theory.
• First isomorphism : subgroup, normal divisor, then applies ${\displaystyle H}$ ${\displaystyle N}$ ${\displaystyle H/(H\cap N)\cong HN/N.}$
• Van Est's theorem : Continuous cohomology of Lie groups
• Euclid's Lemma : If a prime number divides a product of natural numbers, so does one of the factors.
• Euclid's theorem : There are infinitely many prime numbers.
• Euler theorem (also called Euler-Fermat theorem ): Generalization of Fermat's little theorem: ${\displaystyle a^{\varphi (n)}\equiv 1\,(\mathrm {mod} \,n)}$
• Euler's theorem (geometry) : Formula for the distance between the centers of the circumference and the inscribed circle of a triangle ${\displaystyle d}$
• Euler's theorem (prime numbers) : summation of the reciprocal values ​​of the prime numbers
• Euler's theorem (square geometry) : side lengths and diagonals of a square
• Euler - Hierholzer theorem : A connected graph is an Eulerian graph if and only if it only has vertices of even order.
• Euler shear Polyedersatz : For a three-dimensional polyhedron applies corners - Edge + = surfaces. 2

F.

• Fagin's theorem : The set of all sentences that can be described with the aid of the existential predicate logic of the second order is the complexity class NP.
• Flag theorem : characterization of trigonalizable matrices
• Lemma von Farkas : A duality lemma for solving linear systems of inequalities
• Fatou's Lemma : Theorem about the Lebesgue integral of a Limes inferior of a sequence of functions
• Feit - Thompson theorem : Every group of odd order is solvable.
• Set of Pest : set on the convergence of the arithmetic mean of the partial sums of a Fourier series
• Feller's Theorem : Conclusion of the validity of the central limit theorem for the Lindeberg condition
• Little Fermat 's theorem : For every integer and every prime number is . ${\displaystyle a}$ ${\displaystyle p}$ ${\displaystyle a^{p}\equiv a\,(\mathrm {mod} \,p)}$
• Set of Fermat-Wiles Taylor , and Fermat's Last Theorem , or Fermat's Last theorem : For there is no natural numbers with ${\displaystyle n>2}$ ${\displaystyle a,b,c>0}$ ${\displaystyle a^{n}+b^{n}=c^{n}}$
• Fermat's set of polygonal numbers: Representation of a natural number as the sum of polygonal numbers
• Fermat 's prime number theorem : A prime numberis the sum of two squares if and only if it has the form. ${\displaystyle p>2}$ ${\displaystyle p=4n+1}$
• De Finetti's theorem : Representation of interchangeable families of random variables
• Finsler - Hadwiger theorem : Describes properties of two squares with a common corner point.
• Set of Fischer - Riesz : every Hilbert space is isomorphic isometric to a -space. ${\displaystyle \ell ^{2}}$
• Fitting's theorem : The complex product of two nilpotent normal divisors is again a nilpotent normal divisor.
• Fixed point set : List of fixed point sets
• Fixed point theorem for whole functions : Existence of fixed points for chaining whole functions
• Floquet's theorem : on the structure of the fundamental matrices of a homogeneous linear system of ordinary differential equations with periodic coefficient matrix
• Fodor's theorem : Regressive functions on stationary sets must be constant on a stationary subset.
• Fraïssé's theorem : Characterization of the elementary equivalence for a finite set of symbols
• Frank's Lemma : Inequality about the relationship of maxima and integrals of finite families of random variables.
• Representation theorem of Fréchet - Riesz : Representation of the dual space of a Hilbert space over the scalar product.
• Freudenthal 's suspension theorem: Theorem about fundamental groups of suspensions of connected CW complexes
• Friendship phrase : In a room in which two people have exactly one mutual friend, there is one person who is friends with everyone.
• Theorem from Friedberg and Muchnik : There are recursively enumerable Turing degrees between and . ${\displaystyle \mathbf {0} }$ ${\displaystyle \mathbf {0} '}$
• Set of Frobenius : Existence of tangential dimensional foliations to dimensional distributions ${\displaystyle k}$ ${\displaystyle k}$
• Theorem of Frucht : Every group is isomorphic to the automorphism group of a graph.
• Set of Fubini : repatriation of multidimensional integrals to one-dimensional integrals
• Set of Fueter - Pólya : The Cantor polynomials are the only square real polynomials that a bijection convey. ${\displaystyle \mathbb {N} ^{2}\rightarrow \mathbb {N} }$
• Fundamental lemma of homological algebra : Lemma for the continuability of chain homomorphisms
• Fundamental theorem of algebra : Above the complex numbers, every polynomial -th degree has zeros (counted with multiples). ${\displaystyle n}$ ${\displaystyle n}$
• Fundamental theorem of analysis : the derivative of the integral function of a function is the function itself.
• Fundamental theorem of arithmetic : Every natural number greater than one has a prime factorization which is unique except for the order of the factors.
• Fundamental theorem of the calculus of variations : existence of minima of real-valued functionals.
• Pentagon set : A three-dimensional pentagon with lots of equal angles and sides is necessarily a flat geometric structure.
• Five lemma : Lemma from homological algebra (diagram hunt)
• Five-color set : Each map can be colored with five colors so that no two countries with the same color border each other.
• Furstenberg 's x2x3 theorem : The orbits of the irrational numbers under a dynamic system are close together.
• Theorem of Fuss : Transfer of Euler's theorem for triangles to tendon tangent quadrilaterals (formula for the distance between the centers of the circumference and the incircle)
• Theorem of soccer : If a ball is rotated in three-dimensional space any number of times on the square, then there are at least two fixed points on the ball surface.

G

• Gaifman's theorem : If there is a finite relational signature, then every sentence in finite models is logically equivalent to a local sentence. ${\displaystyle \sigma }$ ${\displaystyle L_{I}^{\sigma }}$
• Main theorem of the Galois theory : Relations between subgroups of the Galois group and the intermediate bodies of body extensions
• Gantmacher's theorem : A linear operator between Banach spaces is weakly compact if and only if its adjoint operator is weakly compact.
• Gauss lemma : The content of polynomials in factorial rings is multiplicative.
• Gauss's theorem : Polynomial rings over factorial rings are factorial again.
• Gauss-Bonnet theorem : relationship between curvature and Euler characteristic of a compact, orientable, two-dimensional Riemannian manifold
• Gaussian integral theorem (divergence theorem or Gauss-Ostrogradski's theorem ): The surface integral of a vector function is equal to the volume integral of the divergence.
• Gauss - Lucas theorem : The zeros of the derivative of a polynomial lie in the convex hull of the zeros of the polynomial.
• Gauss-Markow theorem : The least squares estimator is a minimally variant linear unbiased estimator.
• Gelfand - Mazur theorem : A -Banach algebra that is a skew body is isomorphic to . ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {C} }$
• Gelfand - Neumark theorem : Two sets of representation for C * -algebras, commutative and general case
• Set of Gelfond - Schneider : and are algebraic numbers with , , is not rational. Then is transcendent. ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle \alpha \neq 0}$ ${\displaystyle \alpha \neq 1}$ ${\displaystyle \beta }$ ${\displaystyle \alpha ^{\beta }}$
• Gentzen 's main clause (also cut clause ): The cut rule in sequential calculi is redundant.
• Set of Gershgorin : estimation of the amount zeros of polynomials depending on the coefficients
• Strong law of large numbers : Almost certain convergence of the arithmetic mean against the expected value.
• Weak law of large numbers : Stochastic convergence of the arithmetic mean against the expected value.
• Law of large numbers : unites weak and strong law of large numbers
• Set of Girsanow : Transformation of stochastic processes in a standardized Wiener process.
• Gleason's theorem : A finite projective plane in which the diagonal points of any complete quadrilateral are collinear is desarguessian.
• Gleason - Kahane - Żelazko theorem : A characterization of the multiplicative functionals on a complex Banach algebra.
• Gliwenko - Cantelli's theorem (fundamental theorem of statistics): convergence of the empirical distribution function according to probability.
• Gnomon's theorem : Equality of area of ​​certain parallelograms
• Godel 's theorem of incompleteness : Every sufficiently powerful formal system is either contradictory or incomplete.
• Godel's theorem of completeness : For the logic of the first level, syntactic and semantic inference are synonymous.
• Goldstine's theorem : The unit sphere of a Banach space lies weakly - * - close to the unit sphere of the bidual space.
• Goodstein's Theorem : Certain sequences of natural numbers eventually become 0 (regardless of Peano arithmetic).
• Lemma from Goursat : Preparatory lemma for Cauchy's integral theorem, version of the integral theorem for triangles
• Green's theorem : Relationship between surface and curve integral.
• Gronwall -Lemma : An implicit integral inequality is deduced from an explicit inequality.
• Grötzsch's theorem (function theory) : The affine mapping between two rectangles is the quasi-conformational mapping of minimal dilation.
• Grötzsch's theorem (graph theory) : A triangle-free planar graph can be colored with three colors.
• De Gua's theorem : spatial analogue to the Pythagorean theorem

H

• Hadamard's theorem of three circles: Estimation of maximum values ​​of holomorphic functions on three concentric circles
• Complete set of Hadamard : Hadamard gap series are nowhere analytically continued outside the convergence circle.
• Hahn-Banach theorem : Continuous linear functionals on subspaces of locally convex spaces can be extended to the whole space.
• Hales-Jewett theorem : Ramsey theory
• Halmos-Savage theorem : Existence of sufficient σ-algebras under dominated distribution classes.
• Hamilton's theorem : A compact, simply connected, 3-dimensional, Riemannian manifold of strictly positive Ricci curvature is diffeomorphic to the 3-sphere.
• Handshake lemma : The sum of the degrees of all nodes of a graph is equal to twice its number of edges.
• Theorem of Hanner : A topological space covered by a finite number of open, absolute surrounding retracts is again an absolute surrounding retract.
• Harcourt's theorem : Relationship between the area of ​​a triangle and the distances of the corner points from an incircle tangent
• Tauber set of Hardy - Littlewood : A theorem on the asymptotic behavior of the coefficients sum of a power series.
• Hardy and Ramanujan theorem : The number of prime factors of has the normal size . ${\displaystyle n}$ ${\displaystyle \log(\log(n))}$
• Harnack 's principle : Axel Harnack's theorem on the convergence behavior of monotonically increasing sequences of harmonic functions.
• Set of Hartman - Grobman : A dynamic system behaves in the environment of a hyperbolic fixed point as the linearized around this point system.
• Set of Hartman - Wintner : law of the iterated logarithm for iid random variables
• Hartogs lemma : Continuation of a holomorphic function defined in a neighborhood of the edge of a poly cylinder into the whole poly cylinder.
• Hartogs ' continuity theorem : Theorem about the continuation of holomorphic functions (generalization of Hartogs' lemma)
• Set of Hartogs (function theory) : Componentwise holomorphic functions are holomorphic.
• Set of Hartog (set theory) : To each well-ordered set, there is a well-ordered set of larger thickness.
• Main theorem of differential and integral calculus : The derivative of the antiderivative of a function is the function itself.
• Main theorem of local curve theory : Existence of curves for given Frenet curvatures
• Main theorem of projective geometry : In every Pappos projective plane, a projective assignment between straight lines is clearly defined by three pairs of points and pixels.
• Main theorem of Riemannian geometry : Existence law for the Levi-Civita connection
• Law on finitely generated Abelian groups : Every finitely generated Abelian group is isomorphic to a finite direct product of cyclic groups.
• Hausdorff's G _δ theorem : Complete metrizability of sets ${\displaystyle G_{\delta }}$
• Hausdorff's maximal chain theorem : A statement equivalent to the axiom of choice
• Heine's theorem : If a function is continuous in a closed interval, then it is even uniformly continuous there.
• Heine-Borel Theorem : The compact subsets of are exactly the bounded and closed subsets. ${\displaystyle \mathbb {R} ^{n}}$
• Marriage Theorem (also Hall theorem ): A theorem from graph theory about the existence of pairings in bipartite graphs
• Hellinger-Toeplitz theorem : Symmetric operators defined everywhere on a Hilbert space are continuous.
• Set of Helly : The average of a finite family of convex sets in is if and non-empty if the average n of each + 1 sets of non-empty. ${\displaystyle \mathbb {R} ^{n}}$
• Helly's selection theorem : existence of vaguely convergent subsequences of bounded sequences of measures and distribution functions.
• Set of Helly-Bray : From the weak / vague convergence of distribution functions, the weak / follows vague convergence of moderation.
• Helmholtz theorem : Description ofspaces as a direct sum of divergence-free functions and gradient fields ${\displaystyle L^{p}}$
• Set of Henkin : sentence about model characteristics of the Term interpretation
• Hensel 's lemma : Theorem about the factorization of polynomials
• Set of Herbrand Theorem certain about testing the impossibility of logical formulas
• Heron's theorem : Calculation of the triangular area from the lengths of the sides
• Hessenberg's theorem (geometry) : In every projective plane, the Pappos theorem follows the Desargues theorem.
• Hessenberg's theorem (set theory) : Every infinite cardinal number is equal to its square.
• Hewitt-Savage's zero-one law : The interchangeable σ-algebra is a P-trivial σ-algebra.
• Hilbert 's basic theorem : Polynomial rings over Noetherian rings are Noetherian again
• Hilbert 's zero theorem : Existence of zeros of ideals ${\displaystyle {\mathfrak {a}}\subsetneq K[X_{1},\ldots ,X_{n}]}$
• Hilbert scher Syzygiensatz : In modern form: Every finitely generated module over the polynomial ring inindefinite has a free resolution of the length. ${\displaystyle n}$ ${\displaystyle \leq n+1}$
• Hilbert s Theorem 90 : Structure of field extensions with a cyclic Galois group
• Set of Hille - Yosida : theorem on infinitesimal generator of a strongly continuous semigroup
• Hindman's theorem : If an IP-set is a finite union, then one of the union- sets is also an IP-set.
• Hirzebruch's signature theorem : relationship between gender and signature of a manifold. ${\displaystyle L}$
• Hjelmslev's theorem : On the position of certain route centers in plane congruence maps.
• Hoeffding inequality : bound for the probability that a sum of random variables deviates from its expected value by more than a given value
• Height Euclid's theorem : linked height of a right triangle with the Hypotenusenabschnitten:. ${\displaystyle h^{2}=p\cdot q}$
• Hölder's theorem : Every Archimedean ordered group is commutative and isomorphic to a subgroup of real numbers.
• Holditch's theorem : If a chord of constant length traverses a convex curve once, the locus of a point of this chord with the distances and to the end points includes an area that is smaller than that of the original curve. ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle \pi ab}$
• Homomorphism Theorem It establishes a close connection between group homomorphisms and normal divisors as well as vector space homomorphisms and sub-vector spaces.
• Hopf's theorem : For every connected, oriented, closed, differentiable n-manifold the degree of mapping is a homotopy invariant.
• Hopf 's maximal ergodic lemma: auxiliary result of the ergodic theory, which can be used to prove the individual ergodic theorem.
• Set of Hopf - Rinow Characterization of geodesic completeness of connected Riemannian manifolds
• De l'Hospital's theorem : If the limit values ​​of the differentiable functions in the numerator and denominator of a fraction are zero or infinite, then the limit value of this fraction is the same as that with the derivatives of the functions in numerator and denominator.
• Horseshoe lemma : three objects of a short, exact sequence can be resolved projectively or injectively so that the resolutions form an exact sequence.
• Hughes-Piper Theorem : Characterization of strong resolutions of block plans by means of intersection numbers
• Hurwitz's theorem (automorphism groups) : The automorphism group of a hyperbolic, compact, Riemannian surface is finite.
• Hurwitz theorem (theory of functions) : number of zeros of a holomorphic limit function.
• Hurwitz theorem (number theory) : Diophantine approximation of irrational numbers

I.

• Identity theorem for holomorphic functions : A holomorphic function is already determined by its values ​​on a set with an accumulation point.
• Theorem of the Hedgehog : On a sphere there is a tangential, continuous, nowhere vanishing vector field if and only if is odd. ${\displaystyle S^{n}}$ ${\displaystyle n}$
• Theorem of the implicit function : Properties of differentiability in solving implicit equations
• Infinite Monkey Theorem : Of the infinite number of monkeys who happen to type on a typewriter, one almost certainly writes William Shakespeare's Hamlet
• Integral theorem : List of integral theorems
• Interval nesting : Each interval nesting defines a well-defined real number
• Theorem of the invariance of the dimension : If two open sets are out or homeomorphic, then is . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{m}}$ ${\displaystyle n=m}$
• Theorem of the invariance of the area : An injective, continuous mapping of a area in the is a homeomorphism on the image, which is also a area. ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$
• Theorem of the invariance of open sets : An injective, continuous mapping of an open set in the is a homeomorphism on the image, which is also an open set. ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$
• Ionescu-Tulcea theorem : Existence of probability measures on countable product spaces.
• Law of the iterated logarithm : Several limit theorems from stochastics
• Itō's lemma : A basic theorem on stochastic integration

J

• Jacobi's theorem : A theorem about the number of representations of a natural number as a sum of four squares
• Jacobson's theorem of tightness : If M is a true, simple R-module, then every end _R (M) -linear mapping on a finite set behaves like the left multiplication with a ring element.
• Compactness criterion from James : A non-empty, weakly closed subset of a Banach space is exactly weakly compact if every continuous linear functional on it assumes its absolute maximum.
• Theorem of James : A Banach space is reflexive if and only if every continuous linear functional on the unit sphere assumes its norm.
• Japanese theorem for concyclic polygons : The sum of the incircle radii of a triangulated, concyclic polygon is independent of the triangulation chosen.
• Japanese sentence for concyclic quadrilaterals : the centers of the four incircles of a concyclic quadrilateral form a rectangle.
• Jensen's theorem : There are infinitely many irregular prime numbers.
• Jensen's inequality : An elementary inequality for convex and concave functions.
• Jegorow's theorem : A measure theoretical theorem about almost uniform convergence of function sequences
• Jones Lemma : A criterion for the non-normality of a topological space
• Lemma von Jordan : Lemma for the calculation of integrals by means of function theory, integration path = semicircle with increasing radius
• Jordan scher curve theorem : A simply closed continuous curve divides the plane into two areas.
• Jordan-Brouwer decomposition theorem : The complements of homeomorphic compact subsets deshave the same number of path components. ${\displaystyle \mathbb {R} ^{n}}$
• Jordan - Hölder theorem : Any two series of compositions in a group are equivalent. ${\displaystyle G}$
• Jordan - von Neumann theorem : A normalized space that satisfies the parallelogram equation is a Prähilbert space.
• Jung's theorem : Required ball size to capture finitely many points

K

• Theorem of Kadets - Snobar : For a -dimensional subspace of a standardized space there are projections with norm . ${\displaystyle n}$ ${\displaystyle \leq {\sqrt {n}}}$
• Kadison's transitivity theorem: A theorem about the richness of the image of an irreducible representation of a C * algebra
• Fixed point theorem of Kakutani : Closed, convex correspondences on compact, convex sets in have at least one fixed point. ${\displaystyle \mathbb {R} ^{n}}$
• Theorem of Kakutani - Yamabe - Yujobô : For every continuous function of the sphere there is a system of orthonormal vectors with the same function value. ${\displaystyle \mathbb {R} }$
• Kan and Thurston's theorem : For every path-connected space X there is a discrete group π and a homology isomorphism K (π, 1) → X.
• Kantorowitsch's theorem : Sufficient condition for the convergence of Newton's method
• Kaplansky density theorem : If a C * -algebra is dense in a Von-Neumann-algebra in the strong operator topology, then this density relation also applies to the unit spheres.
• Kasparow's stabilization theorem : The Hilbert- module of a C * -algebra absorbs every countably generated Hilbert- module. ${\displaystyle B}$ ${\displaystyle H_{B}}$ ${\displaystyle B}$ ${\displaystyle B}$
• Kasparov's Theorem - Stinespring : A version of Stinespring's Theorem
• Katětov's interpolation theorem : Between an above-semi-continuous and below-semi-continuous function on normal space there is a continuous one.
• Theorem : The square of the height of a right triangle is equal to the product of the hypotenuse sections.
• Kawasaki theorem : Origamis can be folded flat
• Chain rule : Theorem about the derivation of a concatenation of differentiable functions
• Set of Kirchhoff - Trent : set to calculate the number of stands in a graph.
• Class number formula : class number of a number field
• Kleene's fixed point theorem : Every totally calculable function has a fixed point with regard to every Gödel numbering.
• Combination theorem from Klein : Theorem about free products from Klein's groups
• Theorem of the constant chord : invariance of the chord length in the configuration of two intersecting circles
• Set of Knuth : set on the production of pseudo-random numbers using linear congruential
• Kolmogoroff's criterion for normalization: A Hausdorff topological vector space can be normalized if and only if it has a bounded and convex null neighborhood.
• Kolmogorow-Arnold-Moser theorem : Existence of quasi-periodic solutions for a certain class of differential equations
• Kolmogorov-Chentsov theorem : Existence of locally Hölder-continuous modifications of stochastic processes.
• Set of Kolmogorov - Riesz : compactness criteria -spaces ${\displaystyle L^{p}}$
• Kolmogorov's zero-one law : Terminal events of a sequence of independent σ-algebras have the probability 0 or 1.
• Kolmogoroff 's three-series theorem : Criteria for the almost certain convergence of a series of real random variables.
• Kolmogorov's extension theorem : Existence of probability measures on product spaces.
• Compactness theorem (logic) : A set of formulas of the first-level predicate logic can be exactly fulfilled if every finite subset can be fulfilled.
• Kōmura's theorem - Kōmura : characterization of nuclear spaces as subspaces of powers of the space of rapidly falling sequences
• König Lemma : A connected graph with infinitely many vertices of finite degree has an infinitely long path.
• König's theorem (graph theory) : If there are only even circles in a graph, it is bipartite.
• König's theorem (set theory) : A theorem about a strict inequality between cardinal numbers.
• Set of Korowkin : convergence of linear, positive operators on spaces of continuous functions.
• Correspondence theorem (group theory) : The subgroups of a factor group G / N correspond one-to-one to the subgroups of group G, which comprise N.
• Correspondence theorem (stochastics) : Relationship of distribution functions and probability measures on the real numbers.
• Cosines : Generalization of the Pythagorean theorem to general triangles: . ${\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos \gamma }$
• Kerin-Milman's theorem : Convex, compact sets in locally convex spaces are the convex hulls of their extremal points.
• Set of Kerin - Smulian : criterion for the weak - * - isolation of a convex set in the dual space of a Banach space.
• Kronecker's Approximation Theorem: A theorem about the approximation of irrational numbers by rational ones.
• Kronecker 's Lemma : A convergence statement about weighted sums.
• Kronecker's theorem (body theory) : For every non-constant polynomial over a body there is a body expansion in which the polynomial has a zero.
• Kronecker's theorem on series convergence : A criterion for the convergence of series
• Kronecker - Weber theorem : An algebraic number field with an Abelian Galois group is contained in a circle division field.
• Average theorem by Krull : Theorem about powers of ideals and finitely generated modules of a Noetherian ring.
• Theorem of Krull - Remak - Schmidt : Groups or modules with finiteness requirements are the product of indivisible subgroups or sub-modules.
• Kruskal's theorem : The class of finite trees is well-quasi-ordered through the quasi-order relation of the formation of topological minors.
• Kummer's theorem : The Fermat's conjecture is correct as long as the exponent in Fermat's equation is a regular prime number.
• Theorem of Künneth : The homology of a tensor product of chain complexes is equal to the tensor product of the homologies except for torsion.
• Set of Kunugui : Every metric space can be isometrically embedded in a Banach space.
• Kuratowski's theorem : provides a criterion for whether a graph is planar (flattenable) or not.
• Kurepa's theorem : provides a logically equivalent formulation of the axiom of choice in the language of order theory.
• Kurosch subgroup theorem: A sentence about the structure of subgroups of free products

L.

• Ladner's theorem : If there are NP-problems that do not lie in P and are not NP-complete. ${\displaystyle P\not =NP}$
• Set of Lagrange : The order of a subgroup of a finite group divides the group order.
• Lancret's Theorem : Characterization of a slope line by curvature and torsion.
• Set of Lasker-Noether : representation of an ideal as the average primary ideals in Noetherian rings
• Lavrentieff's continuation : A homeomorphism between subspaces of complete, metric spaces can be extended to surrounding G _δ -sets.
• Fixed point theorem Lawvere : fixed-point property in Categories
• Lax's equivalence theorem : In the case of correctly posed linear initial value problems, the finite difference method converges if and only if stability is present.
• Lax-Milgram Lemma : Representation of continuous, coercive sesquilinear forms.
• Decomposition theorem of Lebesgue : Decomposition of a measure in an absolutely continuous and a singular part
• Fixed point theorem of Lefschetz : Number of fixed points of a continuous mapping of a compact topological space on itself
• Set of Lefschetz on hyperplane sections : Topology of hyperplane sections of projective varieties
• Set of Lehmann - Scheffe Theorem about evenly best unbiased estimators.
• Leibniz's Theorem : An equation for affine coordinates in a triangle.
• Leray's theorem : Determination of the sheaf cohomology by means of suitable coverages
• Leray - Schauder -Alternative : A sufficient criterion for the existence of a fixed point
• Lerch's Uniqueness Theorem : Determination of a function by its Laplace transform
• Lerch's theorem : Congruence of power sums
• Lester's theorem : The two Fermat points, the center of the Feuerbach circle and the center of a non-isosceles triangle lie on a circle.
• Levi's theorem : Every finite-dimensional real or complex Lie algebra is the semidirect one of its radical and a semi-simple Lie algebra.
• Inequality by Beppo Levi : Fundamental inequality about distances in (pre-) Hilbert spaces, which leads to the projection theorem.
• Lévy's Theorem : Simple Convergence Theorem for Martingales.
• Arc sine law of Lévy : Distribution of the return time of the Wiener process
• Lévy continuity theorem : relationship between the convergence of probability measures and their characteristic functions.
• Lévy - Khinchin formula : characterization of infinitely divisible probability distributions on the real numbers.
• Lie's theorem : Finite-dimensional, solvable, complex Lie algebras have an invariant flag.
• Lie 's sentences : Three sentences about Lie groups.
• Liebmann's theorem : Characterization of spheres in Euclidean space
• Central limit theorem by Lindeberg-Feller : Generalization of the central limit theorem for non-identically distributed random variables and a certain degree of stochastic dependence.
• Lindeberg theorem : The Lindeberg condition is sufficient for the central limit theorem
• Lindelöf's first theorem : Thickness of the complement of the set of condensation points in topological spaces with countable basis
• Lindelöf's Theorem : Second countable spaces are Lindelöfsch.
• Set of Lindemann-Weierstrass : transcendence of the mathematical constant π.
• Lindenbaum's theorem : Every consistent set of formulas in first-order predicate logic can be expanded to a consistent and complete theory.
• Lindström's Theorems : First order predicate logic is the only logic that satisfies the compactness theorem and the Löwenheim-Skolem theorem.
• Set of Liouville (Differential Geometry) : Formula for calculating the geodetic curvature of surface curves
• Set of Liouville (function theory) : Restricted entire functions are constant.
• Liouville's formula : Relationship between Wronski determinant and trace of the coefficient matrix of a linear differential equation system
• Set of Little Theorem about the average number of customers in a queuing model
• Lyapunov's theorem : The Lyapunov condition is sufficient for the validity of the central limit theorem
• Löb's theorem : A proposition from the logic of provability.
• Theorem of Lochs : Convergence rate of continued fractions
• Set of Łoś : set of first as to the validity of sets of the predicate logic stage in Ultra products
• Plumb line set : angle between plumb lines
• Lovász -Local Lemma : A condition that an average of events has a positive probability.
• Lovász-Stein's theorem : Relationship of the hyper-edges required to cover a hypergraph to other parameters of this hypergraph.
• Löwenheim-Skolem theorem : Satisfiable, countable sets of statements of the first-order predicate logic have countable models.
• Lumer - Phillips theorem : Characterization of the contraction half-groups
• Lusin's theorem : Relationship between continuous and measurable functions

M.

• Set of Mackey : theorem on bounded sets in locally convex spaces
• Set of Mackey - Arens : Set on allowable topologies on a locally convex space.
• Maekawa's sentence : In a flat folded figure (origami), the number of mountain and valley folds differs by exactly two
• Majorant criterion : A series with absolutely convergent majorants is itself absolutely convergent.
• Theorem about the majorized convergence : Theorem about the interchangeability of integration and point-wise convergence of the integrands
• Malcev's theorem : Every finitely generated subgroup of is residual finite. ${\displaystyle GL(n,\mathbb {C} )}$
• Set of Malgrange - Honorary Award : linear partial differential equations with constant coefficients have a Green's function.
• Marczewski - Szpilrajn theorem : Every partial order can be expanded to a linear order.
• Marden's Theorem : A relationship between the zeros of a third line polynomial and the zeros of its derivative
• Lemma von Margulis : Theorem about the fundamental group of the thin part of a complete Riemannian manifold with intersection curvatures in [-1,0].
• Markow's theorem : Conditions for the equivalence of entanglements obtained as the end of a braid
• Markov's inequality : upper bound for the derivation of polynomials.
• Set of Marsaglia : set on pseudo-random numbers from linear congruential
• Martingale convergence theorem : A submartingale with upwardly limited expectation values ​​of the positive parts converges almost certainly to a function. ${\displaystyle L^{1}}$
• Set of Maschke : decomposition of a group display in a direct Summer irreducible representations
• Maskit's combination theorem : Discreetness of amalgamated products from small groups
• Matsumoto's theorem : Description of the group of a body ${\displaystyle K_{2}}$
• Max-Flow-Min-Cut-Theorem : The maximum flow in the network has exactly the value of its minimum cut.
• Mazur's theorem : Suitable convex combinations of terms of weakly convergent sequences are strongly convergent (functional analysis).
• Mazurkiewicz's theorem : The plane contains a subset that has exactly two points in common with every straight line.
• McShane's lemma : Continuability of Lipschitz continuous functions on subspaces of metric spaces.
• Menelaus theorem : a statement about straight lines that intersect triangles (product of partial ratios)
• Menger's theorem : The maximum number of edge-disjoint paths is equal to the minimum separating edge set.
• Set of Menger - Nöbeling : embedding finite Kompakta in . ${\displaystyle \mathbb {R} ^{n}}$
• Mercer's theorem : Representation of the continuous kernel of a positive integral operator over the unit square as a uniformly convergent series.
• Set of Mergelyan : If K is a compact subset without holes in , the polynomials are close in . ${\displaystyle \mathbb {C} }$ ${\displaystyle A(K)}$
• Mertens theorem (Cauchy product) : Theorem about the convergence of a Cauchy product of two series
• Mertens theorem (resultant system) : A theorem about common zeros of homogeneous polynomials
• Mertens' theorem (number theory) : A theorem about the asymptotic behavior of the series over the reciprocal values ​​of the prime numbers
• Meyers - Serrin theorem : The functions that can be differentiated as often as you like lie close together in the Sobolew spaces.
• Milman's Theorem : Uniformly convex spaces are reflexive.
• Milnor - Moore theorem : A relationship between a Hopf algebra and its Lie algebra of the primitive elements
• Milnor - Thom theorem : Estimation of the number of connected components of the set of roots of a polynomial.
• Minkowski's theorem : A compact, convex set in a finite-dimensional space is the convex hull of its extremal points.
• Minkowski inequality : triangle inequality inspaces ${\displaystyle L^{p}}$
• Minkowski lattice point theorem : statement about the density of lattice points
• Minty - Browder's theorem : Monotonic, coercive, hemistic operators of a separable, real Banach space in its dual space are surjective.
• Miquel's theorem : Three circles, each running through a corner and the two adjacent sides of a triangle, have a common point of intersection
• Mitchell's embedding theorem : Abelian categories can be embedded in concrete categories of links modules over a ring.
• Mittag - Leffler theorem : Existence of meromorphic functions at given poles
• Mean value theorem of differential calculus : Between every two digits of a differentiable function there is a point with a tangent slope = secant slope
• Mean value theorem of integral calculus : Formula for the integral of a product of functions by means of a mean value of a function
• Möbius-Pompeiu's theorem : property of equilateral triangles
• Theorem of modularity : Relationship between elliptical curves and modular shapes (formerly Taniyama-Shimura conjecture)
• Mohr's sentence - Mascheroni : Every construction with compass and ruler can already be carried out with the compass alone.
• Set of Moivre - Laplace : convergence of the binomial distribution to the normal distribution
• Moivre 's sentence : ${\displaystyle \left(\cos x+i\,\sin x\right)^{n}=\cos \left(n\,x\right)+i\,\sin \left(n\,x\right)}$
• Monge's theorem : A theorem about a specific point on a tetrahedron
• Theorem about monotonic classes : Generation of limited measurable functions from multiplicative classes of limited measurable functions
• Theorem of monotonic convergence : interchangeability of integration and point-wise, monotonic limit
• Monotony criterion : A monotonic sequence of real numbers converges to a limit if and only if it is bounded.
• Montel's theorem : A locally uniformly bounded sequence of holomorphic functions has a compactly convergent subsequence.
• Mordell's Theorem - Weil For an Abelian variety over a number field , the group of -rational points is finitely generated. ${\displaystyle K}$ ${\displaystyle K}$
• Morera's theorem : If the integral of a continuous function vanishes over all triangle edges, then it is holomorphic.
• Morita theorem : Every regular Lindelöf room is paracompact.
• Morley's Theorem (Geometry) : The Morley triangle of a triangle is always equilateral.
• Morley's Theorem (Logic) : Theorem on the categoricity of countable theories of first-order predicate logic
• Mostow's theorem : Every connected algebraic group over a field of characteristic 0 has a Levi decomposition.
• Mostow - Prasad Rigidity Theorem : Rigidity of hyperbolic metrics
• Mostowski collapse : A sentence from set theory about well-founded and extensional relations
• Mourier's theorem : Sufficient condition for the validity of the strong law of large numbers in separable Banach spaces.
• Myers - Steenrod theorem : The isometric group of a complete Riemannian manifold is a Lie group.

N

• Naimark's theorem : Positive, 1-bounded operators on commutative C * -algebras are compressions of homomorphisms.
• Lemma of Nakayama : If there is a finitely generated module and an ideal contained in the Jacobson radical , then is . ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle {\mathfrak {a}}M=M}$ ${\displaystyle M=0}$
• Napoleon's theorem : The triangle formed by the centers of gravity of the equilateral triangles over the sides of a triangle is equilateral.
• Nash embedding theorem : Riemannian manifolds can be isometrically embedded in a Euclidean space . ${\displaystyle \mathbb {R} ^{n}}$
• Nash's Theorem : Existence of negotiation solutions (game theory)
• Secondary angle set : Secondary angles add up to 180 °.
• Neunerlemma : Diagram Hunting in a Diagram . ${\displaystyle 3\times 3}$
• Newton's theorem : The inscribed center of a tangent quadrilateral lies on the Newton straight line.
• Neyman - Pearson- Lemma : The Neyman-Pearson test is an equally best test
• Set of Nielsen - Schreier : subgroups of free groups are free.
• Noether 's normalization theorem : A finitely generated algebra over a field is finite over a polynomial ring.
• Nordhaus - Gaddum theorem : Inequalities for the sum and product of the chromatic numbers of a finite graph and its complementary graph
• Zero sequence criterion : If the sequence of the summands of a series does not form a zero sequence, then the series diverges.

O

• Theorem about the open mapping : Continuous, linear, surjective mappings between Banach spaces are open.
• Openness theorem : Non-constant holomorphic functions are open.
• Set of Oka : theorem on the approximation of analytic functions by polynomials in multiple variables
• Olivier's theorem : If the series is monotonically decreasing and the series converges, then it is a null sequence. ${\displaystyle (a_{n})_{n}}$ ${\displaystyle (n\cdot a_{n})_{n}}$
• Optional sampling theorem : In a fair game, a stop time cannot improve the payout.
• Optional stopping theorem : stopped (sub- / super) martingales are again (sub- / super) martingales.
• Orlicz - Pettis' theorem : A weak partial series convergent series in a Banach space is also partial series convergent with regard to the norm topology.
• Osgood's Lemma : A continuous function holomorphic in every variable is already holomorphic.
• Osgood's theorem (function theory) : Injective, holomorphic functions are biholomorphic.
• Osgood's theorem (functional analysis) : A pointwise upwardly bounded family of subcontinuous functions is equally upwardly bounded on an open set.
• Set of Ostrowski : a non-trivial amount function is equivalent to the Euclidean or a p-adic amount. ${\displaystyle \mathbb {Q} }$
• Inequality Ottaviani Skorokhod : inequality over finite families of stochastically independent random variables.

P

• Paley's Theorem : Existence of Hadamard Block Plans.
• Paley - Wiener theorem : Characterization of the Fourier-Laplace transformations of smooth functions or temperature-controlled distributions with compact support by means of growth conditions
• Paley - Wiener - Zygmund theorem : Path properties of the Wiener process.
• Palm - Chinchin's theorem : The superposition of renewal processes asymptotically approaches a Poisson process if the events occur relatively rarely.
• Pappos' theorem : If the corner points of a hexagon lie alternately on two straight lines, the intersection points of opposite sides also lie on a straight line.
• Area formula of Pappus : areas of parallelograms over triangle sides
• Parseval's equation : Equation in Hilbert spaces that represents the norm of a vector using an orthonormal basis.
• Pascal's theorem : If the corner points of a hexagon lie on a conic section, the intersection points of the three opposite pairs of sides of the hexagon lie on a straight line.
• Peano's existence theorem : existence theorem from the theory of ordinary differential equations (continuous case)
• Decomposition method by Pelczynski : A theorem for the construction of isomorphisms between Banach spaces
• weak perfect graph set : A graph is perfect if and only if its complementary graph is perfect.
• Set of Perron - Frobenius : A matrix with positive components has the spectral radius as an intrinsic value plus an eigenvector with positive components.
• Set of Peter - Weyl : theorem on the Fourier transformation of a compact group
• Pettis' measurability theorem : A characterization of measurable Banach space valued functions.
• Picard's theorem : The picture of a non-constant whole function is whole with at most one exception point ${\displaystyle \mathbb {C} }$
• Picard-Lindelöf's theorem : Existence and uniqueness theorem for ordinary differential equations (Lipschitz continuous case)
• Set of pick Let the area of the polygon, the number of lattice points inside the polygon and the number of lattice points on the boundary of the polygon, then: . ${\displaystyle A}$ ${\displaystyle I}$ ${\displaystyle R}$ ${\displaystyle \textstyle A=I+{\frac {R}{2}}-1}$
• Pitot's theorem : In a tangent quadrilateral the two sums of the lengths of opposite sides are equal.
• Pitt theorem : For every continuous linear operator is compact. ${\displaystyle 1\leq p<r<\infty }$ ${\displaystyle \ell ^{r}\rightarrow \ell ^{p}}$
• Pizza theorem : Theorem about breaking a circle into equal parts.
• Set of Plancherel : The Fourier transform gives an isometric between Hilbert spaces.
• Platonow's Theorem : A theorem about virtually residual p-finite subsets of the general linear group.
• Poincaré- Lemma : Closed differential forms in star-shaped areas are exact.
• Poincaré-Bendixson theorem : A theorem about the behavior of trajectories in two-dimensional continuous dynamic systems.
• Set of Poincaré - Birkhoff - Witt : set on the basis of the universal enveloping Lie algebra
• Poincaré-Bohl's theorem : Statement about the Brouwer's degrees of mapping of two continuous vector fields.
• Set of Poincaré-Hopf : This sentence shows a relationship between zeros of a vector field and the Euler characteristic of the underlying surface.
• Set of Poincaré - Volterra : A sentence on retransfer topological properties through open continuous maps
• Poincaré conjecture (proven by Perelmann): Every simply connected, compact, unbounded, 3-dimensional manifold is homeomorphic to the 3-sphere.
• Set of Polya : Is continuous, straight, convex on with , it is the characteristic function of a probability measure. ${\displaystyle f\colon \mathbb {R} \to [0,1]}$ ${\displaystyle [0,\infty )}$ ${\displaystyle f(0)=1}$ ${\displaystyle f}$
• Set of Pólya : recurrence and transience symmetric simple random walks.
• Poncelet's closure theorem : existence of an infinite number of corners which are related to conic sections. ${\displaystyle n}$
• Poncelet-Steiner's theorem : Every construction with compass and ruler can also only be carried out with a ruler, provided that a solid circle and its center are given.
• Duality theorem of Pontryagin : Canonical isomorphism of locally compact Abelian group for their Bidualgruppe
• Portmanteau Theorem : A Characterization of Convergence in Distribution of Random Variables
• Pratt's theorem : Conditions for the interchangeability of integration and limit value formation of a function sequence when constricted by locally convergent function sequences.
• Theorem of the primitive element : Every finite, separable body extension is simple.
• Prime number theorem : Theorem on the asymptotic density of prime numbers: ${\displaystyle \pi (x)\,\sim \,x/\ln(x)}$
• Convergence criterion of Pringsheim : convergence criterion for continued fractions
• Principle of local reflexivity : the bidual of a Banach space can be represented finitely in this.
• Set of Prokhorov : Firm, limited amounts of measurements are relatively weakly sequentially compact.
• Product rule : Theorem about the derivation of a product of differentiable functions
• Projection theorem : For every closed subspace of a Hilbert space there is an orthogonal projection.
• Projection set (triangle) : For two sides of a triangle, the rectangles from one side and the projection of the other side have the same area.
• Ptolemy's theorem : In a square chord, the product of the diagonals is equal to the sum of the products of the opposite sides.
• Pugh's closure lemma : approximation of a dynamic system with non-moving points by dynamic systems with periodic orbits.
• Set of Pythagoras : Relationship between the three side lengths of a right triangle . ${\displaystyle a^{2}+b^{2}=c^{2}}$

Q

• Quadratic reciprocity law : Theorem for calculating the Legendre symbol
• Quotient criterion: convergence criterion for series
• Quotient rule : Theorem about the derivation of a quotient of differentiable functions

R.

• Raabe's criterion : A convergence criterion for infinite series
• Rademacher's theorem : Lipschitz continuous functions can be differentiated almost everywhere.
• Rado's selection principle : A sentence about continuations of given finite selection functions
• Radon's theorem : division of -sets in -dimensional space so that the convex hulls of the parts intersect ${\displaystyle (n+2)}$ ${\displaystyle n}$
• Radon-Nikodym theorem : Existence of densities with respect to a measure
• Set of Radon-Riesz : Relationship of standard convergence and weak convergence in -spaces. ${\displaystyle {\mathcal {L}}^{p}}$
• Ramanujan conjecture : Estimation of the Fourier coefficients of the discriminant.
• Ramsey's theorem : existence of monochromatic subgraphs
• Ramsey theorem (set theory) : existence of homogeneous subsets in decompositions of the set of all n-element subsets of a countable set
• Rank sentence : Dimension formula in finite-dimensional vector spaces: . ${\displaystyle \dim V=\dim \operatorname {Kern} (f)+\dim \operatorname {Bild} (f)}$
• Rao - Blackwell Theorem : Criterion for Best Estimators (Mathematical Statistics)
• Lemma von Rasiowa - Sikorski : Existence of generic filters for countable systems of order-dense sets on quasi-orders.
• Theorem about rational zeros : properties and computation of rational zeros of a polynomial
• Rédei's theorem : In a finite tournament with at least two nodes, the number of Hamiltonian orbits in it is always an odd number.
• Reflection principle (set theory) : For every sentence of ZF set theory there is already a set that reflects it.
• Reflection principle (Wiener process) : A Wiener process mirrored at a stopping time is again a Wiener process.
• Theorem of regular value : level sets of regular values ​​are submanifolds
• Reiman's Theorem : An estimate of the number of edges in a finite, simple graph without four circles
• Recursion theorem : Every total calculable function has a fixed point with regard to every Godel numbering.
• Residual theorem : Fundamental theorem of function theory for the computation of integrals
• Reynolds ' transport theorem: Theorem from continuum mechanics
• Set of de Rham : The De Rham cohomology of compact orientable manifolds is naturally isomorphic to the singular cohomology with real coefficients.
• Richardson's Theorem : Every finite directed graph without circles of odd length has at least one kernel.
• Formula by Riemann - Hurwitz : Relationship between branching order, number of leaves and gender in holomorphic images of compact Riemann surfaces.
• Set of Riemann - Roch : number of linearly independent meromorphic functions with predetermined zeros and poles on compact Riemannian surfaces
• Riemann 's illustration theorem : biholomorphism classification of simply connected Riemann surfaces.
• Riemann 's theorem of liftability : Bounded singularities of holomorphic functions can be lifted.
• Riemann 's rearrangement theorem : Theorem about the rearrangement of real series
• Lemma von Riemann - Lebesgue : The Fourier transforms of functions vanish at infinity. ${\displaystyle L^{1}}$
• Representation theorem from Riesz - Markow : Representation of positive linear forms through Radon measures or the representation of the corresponding dual spaces resulting from this.
• Riesz's theorem of compactness : characterization of finite-dimensional vector spaces
• Lemma von Riesz : Theorem about closed subspaces in normalized spaces
• Complete set of Riesz : completeness of -Spaces ${\displaystyle L^{p}}$
• Ringel - Youngs' theorem : Necessary number of colors to color a graph on a surface given gender
• Rochlin's Lemma : Partitioning of the phase space of a dynamic system.
• Theorem of Rolle : Every continuous and differentiable function has a horizontal tangent between two zeros at at least one point.
• Rosser's theorem : There is a lower bound for the nth prime number : ${\displaystyle p_{n}}$ ${\displaystyle p_{n}>n\cdot \ln n.}$
• Roth's theorem : In a subset of the whole numbers with positive upper density there are infinitely many arithmetic sequences of length 3.
• Rouché's theorem : Comparison of the number of zeros of two holomorphic functions.
• Routh's theorem for area calculation in triangles.
• Rückert's basic theorem : The ring of convergent complex polynomials inindeterminates is Noetherian. ${\displaystyle n}$
• Runge 's approximation theorem : Approximation of holomorphic functions by polynomials and rational functions
• Russo Theorem - Dye : In a C * -algebra with a unitary element, the unit sphere is equal to the standard closure of the convex hull of the unitary elements.
• Fixed point theorem by Ryll-Nardzewski : Every semigroup of weakly continuous affine isometries of a weakly compact convex set has a fixed point.
• Ryll-Nardzewski Theorem : Characterization of -categorical theories ${\displaystyle \omega }$

S.

• Sard's theorem : The set of critical values ​​of a sufficiently often differentiable mapping between two manifolds has the Lebesgue measure 0
• Sarkovskii theorem : Number of possible periods in the iteration of a continuous function
• Sárkőzy's Theorem : Proof of Erdős' square-freedom conjecture for large numbers.
• Schanuel's Lemma : A lemma from homological algebra about projective resolutions
• Fixed point theorem of Schauder : Existence of fixed points of continuous functions on convex, compact sets
• Schauder's theorem : A linear operator between Banach spaces is compact if and only if its adjoint operator is compact.
• Set of Scheffe : Convergence in th middle ${\displaystyle p}$
• Leg transversal theorem: Theorem of elementary geometry about length relationships to transversals in isosceles triangles, which is equivalent to the Pythagorean theorem
• Scherk's theorem : Every prime number can be obtained by adding and subtracting the preceding prime numbers and the 1.
• Schilow 's idempotent theorem : Existence of idempotent elements in commutative Banach algebras
• Serpent lemma : provides link homomorphisms for long exact sequences
• Schoen assumption : continuation of quasi-conformal mapping of the 2-sphere to the 3-ball.
• Schoenflies Theorem : A homeomorphism between a closed Jordan curve and the unit circle can be continued on the plane.
• Van Schooten's theorem : In an equilateral triangle, the distance from a perimeter to one of the corners is equal to the sum of the distances to the other two corners.
• Barrier Lemma : In a vector space with a generating system of elements, vectors are linearly dependent. ${\displaystyle n}$ ${\displaystyle n+1}$
• Schreier's theorem : Two normal series of a group G can be extended to equivalent normal series by refinement.
• Schur's Lemma : Theorem about commutators in irreducible representations
• Schur's theorem : An at least partial coloring of the plane is possible with any coloring of the pos. whole numbers with always possible. ${\displaystyle x+y=z}$ ${\displaystyle x,y,z\in \mathbb {Z} ^{+}}$
• Schur - Zassenhaus theorem : On the representability of a finite group as a semi-direct product.
• Schützenberger's theorem : A necessary condition for the existence of certain symmetrical block plans.
• Schwartz's core theorem : A theorem about integral kernels from distribution theory.
• Schwarz's theorem : For functions that are twice continuously differentiable, the order of the derivatives does not matter.
• Schwarz 's lemma : Inequality for holomorphic endomorphisms of the unit circle
• Schwarz 's reflection principle : holomorphism of functions generated by reflection
• Lemma von Schwarz - Pick : Generalization of the Lemma von Schwarz
• Set of Scorza Dragoni : theorem on the solvability of real boundary value problems
• Segre's theorem (Diophantine approximation) : A theorem about the approximation quality of irrational numbers by rational numbers.
• Segre's Theorem (Projective Geometry) : Every oval in a finite Desarguean plane of odd order is a conic section.
• String set : Relationship between intersecting chords of a circle.
• Set of Seifert and van Kampen : theorem on the fundamental group of a topological space
• Lemma von Selberg : Every finitely generated subgroup of , bodies of characteristic 0, is virtually torsion-free. ${\displaystyle GL(n,K)}$ ${\displaystyle K}$
• Silver's theorem : The smallest cardinal number for which the continuum hypothesis is violated cannot be singular with uncountable cofinality.
• Simson's straight line : The base points of a circumference point of a triangle lie on a straight line, which characterizes the circumference points.
• Law of sines : sides and opposite angles in the triangle
• Skorochod representation : A relationship between convergence after distribution and almost certain convergence
• Skorochod 's embedding theorem : embedding random variables in the Wiener process
• Slutsky's theorem : A theorem about the convergence in probability of random variables.
• Sobolew's embedding theorem  : Theorem about compact embedding of Sobolew spaces
• Solovay's theorem
• Spectral theorem : Spectral representation of normal operators.
• Spectral mapping theorem : With some functional calculi, the formation of the spectrum and the insertion into functions can be interchanged.
• Set of Sperner : Anti-chain in the power amount of a n-element set has at most the length over ${\displaystyle n}$ ${\displaystyle {\lfloor {n/2}\rfloor }}$
• Theorem of spheres : An n-dimensional, compact, simply connected, Riemannian manifold with sectional curvature from is homeomorphic to the sphere. ${\displaystyle ]1/4,1]}$
• Stallings theorem : characterization of finitely generated groups with more than one end.
• Stein Lemma : Convergence speed of the error of the 2nd kind in the Neyman-Pearson test
• Steiner's Theorem : Characterization of undeveloped conic sections
• Steiner - Lehmus theorem : If two bisectors in a triangle are of equal length, then it is isosceles.
• Steinhaus's theorem : The set of differences of a Lebesgue measurable set of positive measure is a zero neighborhood.
• Exchange lemma from Steinitz : Lemma for equality of bases of finite-dimensional vector spaces.
• Steinitz theorem : A finite simple graph has a straight line representation as a 3-dimensional polyhedron graph if and only if it is flattenable and 3-fold connected.
• Steinitz shear rearrangement theorem : Theorem about the rearrangement of rows in the ${\displaystyle {\mathbb {R} }^{m}}$
• Stewart's Theorem : Length of a segment from a corner of a triangle to a point on the opposite side
• Set of Stine Spring : Fully positive, 1-bounded operators in C * algebras are compressions of homomorphisms.
• Stirling formula : Asymptotic formula for faculties
• Stokes' theorem : (Generalization of the Gaussian integral theorem)
• Theorem of Stolz : The existence of the limit value of a quotient of two sequences follows from the existence of the limit value of the quotient of the difference sequences
• Stone's theorem : A unitary group is generated by i-times a self-adjoint operator.
• Approximation of Stone-Weierstrass : approximation of continuous functions by polynomials
• Perturbation lemma : Small perturbations of a regular matrix lead to a regular matrix again.
• Theorem of rays : With two rays emanating from the same point and intersecting parallel straight lines, every two sections on one ray behave like the corresponding sections on the other ray; the lines cut out on the parallels behave like the lines measured from the apex on the rays.
• Dispersion Decomposition Theorem : Describes a decomposition of the total sum of squares into an explained sum of squares and a residual sum of squares.
• Step angle set : If two parallel straight lines , and of a third straight line to be cut, so the steps occurring angles are equal. ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$
• Sturm's rule : The number of different zeros of a real polynomial in an interval is equal to the difference between the sign changes in the two Sturm chains at the interval boundaries.
• Sullivan 's theorem of rigidity : Rigidity of quasi- conformal images
• Sylow Sentences : Three sentences aboutsubgroups ${\displaystyle p}$
• Sylvester's theorem of inertia : the number of negative, positive and zero eigenvalues ​​of a symmetric matrix does not depend on the choice of the base of the vector space.
• Sylvester-Gallai's Theorem : For a finite, non-collinear set of points, there is a straight line that goes through exactly two of the points.
• Synge's theorem : Straight-dimensional, orientable manifolds of positive sectional curvature are simply connected.
• Synge - Weinstein's theorem : Orientation-maintaining isometries on straight-dimensional, orientable, Riemannian manifolds with strictly positive intersection curvature have a fixed point.
• Set of Szemerédi : Ramsey, generalizes the set of Van der Waerden.

T

• Takai's duality theorem : The dual cross product to the cross product of a C * -dynamic system is the tensor product with the compact operators.
• Tamano's theorem : A completely regular Hausdorff space is paracompact if and only if the product with its Stone-Čech compactification is normal.
• Tangent set : sides and half angle sums in the triangle
• Theorem of the tangent square: Every square in which the sums of the opposite sides are equal has an inscribed circle and is therefore a tangent square.
• Fixed point theorem of Tarski and Knaster : Existence of fixed points of monotonic mappings on complete lattices
• Taylor's theorem : Every function that is continuously differentiable on a real interval can be expressed by a corresponding Taylor polynomial and a suitable remainder term.
• Lemma von Teichmüller - Tukey : A non-empty set of finite character has a maximum element with respect to the set inclusion.
• Tennenbaum's theorem : No countable non-standard model of Peano arithmetic is computable.
• Set of Thabit : set for befriended numbers construction
• Thales's theorem : For given points , the points that make up a right triangle are exactly the points of the circle around the midpoint of the line . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle C}$ ${\displaystyle ABC}$ ${\displaystyle AB}$
• Theorema egregium : The Gaussian curvature depends only on the coefficients of the first fundamental form of a surface.
• Theorema elegantissimum : The total curvature of a simply connected geodesic triangle is equal to its excess angle.
• Thomsen's theorem : A theorem from triangular geometry about certain closed lines
• Set of Thue - Siegel - Roth : approximation algebraic numbers by rational numbers
• Thurston's theorem : A closed smooth n-dimensional manifold has a smooth -dimensional foliation if and only if its Euler characteristic is zero. ${\displaystyle (n-1)}$
• Set of Thurston-Bonahon : set to dichotomy between geometric finite and infinite geometric surfaces in hyperbolic 3-manifolds
• Continuation of Tietze's theorem : Continuous functions on closed sets of normal spaces can be continued continuously over the entire space.
• Torus theorem : JSJ decomposition of toroidal 3-manifolds.
• Set of Toponogow : are in a manifold with limited upward curvature triangles no thicker than in the same space of constant curvature.
• Theorem of Trachtenbrot : The finite, generally valid first-level propositions cannot be enumerated.
• Transformation theorem: The behavior of integrals under coordinate transformations
• Transversality theorem : approximability of mappings through mappings transversely to a submanifold
• Separation theorem : Separation of convex sets by hyperplanes
• Chebotaryov density theorem: prime numbers in arithmetic progressions on Galois extensions of number fields
• Chebyshev's theorem : Test for elementary integrability of binomial integrals
• Chebyshev -Ungleichung : A random variable deviates likely maximum variance /by more thanthe expected value. ${\displaystyle k^{2}}$ ${\displaystyle k}$
• Tunnell's theorem : Conditions for the congruence of numbers
• Turán's theorem : Determination of the maximum number of edges a graph can have without containing them as a subgraph. ${\displaystyle K_{m}}$
• Tutte's theorem : Characterization of a graph with perfect matching
• Tychonoff's theorem : A product of compact spaces is compact again.

U

• Ugly Duckling Theorem : A sentence from pattern recognition
• Set of Ulam : Each Borel measure on a Polish space is regular and moderate.
• Envelope theorem: A theorem about the behavior of the optimal value of the objective function of a parameterized optimization problem when the parameters change
• Circulation theorem : A twice continuously differentiable, simply closed, regular curve has the rotation number ± 1.
• Theorem about the inverse function : Existence of local inverse functions with an invertible Jacobi matrix
• Universal coefficient theorem : relationship of homology with coefficients in an Abelian group to homology with coefficients in ${\displaystyle \mathbb {Z} }$
• Urysohn's Lemma : Two disjoint, closed sets of a normal space can be separated by a continuous function.

V

• Van Aubel's theorem : The centers of the four squares over the sides of a square are the corners of an orthodiagonal square with diagonals of equal length.
• Set of Van der Waerden : set of combinatorial or Ramsey
• Set of Vantieghem : A number n is prime, if the product of the first Mersenne numbers are congruent is modulo of th Mersenne number. ${\displaystyle n-1}$ ${\displaystyle n}$ ${\displaystyle n}$
• Varignon's theorem : If you connect the centers of neighboring sides of a square, you get a parallelogram.
• Vaught criterion : Categorical theories without finite models are complete.
• Vaught's theorem (maximality principle) : Theorem about a maximality principle equivalent to the axiom of choice
• Theorem of comparability : Every two quantities are comparable with regard to their thickness
• Bonding lemma : Construction of continuous functions through continuous functions that are defined on subspaces
• Displacement theorem : Calculation rule for determining the sum of squared deviations
• Vidav - Palmer theorem : A complex Banach algebra with an involution is a C * -algebra if and only if holds for all . ${\displaystyle A}$ ${\displaystyle \ast }$ ${\displaystyle \|a^{*}a\|=\|a^{*}\|\|a\|}$ ${\displaystyle a\in A}$
• Four-color set : four colors are sufficient to color a map (without ex- or enclaves), so that every two neighboring countries get different colors.
• Four-squares theorem : Any natural number can be written as the sum of four square numbers.
• Four-apex theorem : The curvature function of a simply closed, smooth, flat curve has at least four extreme points.
• Theorem of '' 'Vieta' '' : Relationship between the coefficients and zeros of a quadratic equation
• Vieta's law of roots : The coefficients of a complex polynomial are elementary symmetric functions of the roots .
• Vitali's convergence theorem : criteria under which convergence in the p-th mean and convergence locally to measure agree.
• Set of Vitali Theorem for existence is not Lebesgue measurable quantities.
• Set of Vitali : set for compact convergence of a sequence of holomorphic functions.
• Vitali's coverage theorem : coverage of a set of finite outer Lebesgue measures by elements of a Vitali coverage.
• Theorem of Vitali - Hahn - Saks : The quantitative limit value of a sequence of signed measures is again a signed measure.
• Vizing's Theorem : Estimating the chromatic index of a graph.

W.

• Theorem of Wagner and Fáry : Every plane graph can be converted into a segment graph through a homeomorphism of the Euclidean plane.
• Wald's formula : to calculate the expected value of sums of random quantities with a random index.
• Wallace's Theorem : A product of compact sets in an open set lies in a product of open sets contained therein.
• Wedderburn's theorem : Finite skew bodies are commutative.
• Weierstrass-Casorati theorem : An analytic function comes as close as desired to any complex number in the vicinity of an isolated essential singularity.
• Weierstrass division theorem : In the ring of convergent power series in indeterminates, every element of a Weierstrass polynomial can be divided with a remainder. ${\displaystyle n}$
• Weierstraß's theorem of convergence : A locally uniform limit of holomorphic functions is again holomorphic.
• Weierstrass 's majorant criterion : a criterion for demonstrating uniform and absolute convergence of a series of functions
• Weierstrass 's product theorem : Existence of holomorphic functions for given zero distributions
• Weierstrass preparatory theorem : In the ring of convergent power series inindeterminates, every regular element is a product of a unit and a Weierstrass polynomial. ${\displaystyle n}$
• Because sheer rigidity theorem : Local rigidity of lattices in Lie groups
• Because - conjectures : theorems about the local zeta functions of algebraic varieties
• Wermer's maximality theorem : Disk algebra is a maximal Banach algebra in the algebra of continuous functions on the circular line.
• Set of Weyl on equal distribution : Is irrational, so the result is equally distributed asymptotically. ${\displaystyle 0<y<1}$ ${\displaystyle (ny-\lfloor ny\rfloor )_{n}}$
• Weyl's Theorem : Finite-dimensional representations of semi-simple, finite-dimensional, complex Lie algebras are completely reducible.
• Whitehead's Lemma : The commutator group of the infinite-dimensional linear group over a ring is generated by the elementary matrices.
• Whitehead's Theorem : A weak equivalence between related CW complexes is a homotopy equivalence.
• Set of Whitehead - Serre : A set of the relationship rational homotopy and rational homology groups in simply connected spaces.
• Whitney's embedding theorem : Every -dimensional differentiable manifold that satisfies the second axiom of countability has a closed embedding in . ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{2n}}$
• Theorem of contradiction : a statement cannot be true at the same time as its opposite.
• Set of Wielandt : Characterization of the Gamma function using the functional equation and limitations on the condition of certain strip ${\displaystyle [1,2]}$
• Set of Wiener - Ikehara : set certain about the asymptotic behavior of the coefficients of Dirichlet series
• Wiener - Chintschin theorem : The spectral power density of a stationary random process is the Fourier transformation of the corresponding autocorrelation functions.
• Wilson's theorem : is a prime number if and only if is divisible by . ${\displaystyle p>1}$ ${\displaystyle (p-1)!+1\,\!}$ ${\displaystyle p}$
• Angle bisector theorem (triangle) : The angle bisector in a triangle divides the side opposite the angle in the ratio of the two sides adjacent to the angle.
• Set of Vinogradov : Sufficiently large odd numbers are the sum of three primes.
• Theorem of Wintner - Wielandt : Theorem on the question of the boundedness of the quantum mechanical basic operators.
• Well -ordered theorem : Any amount can be well-ordered.
• Wolstenholme's theorem : If p is a prime number, then the numerator is divisible by. ${\displaystyle 1+1/2^{2}+\ldots +1/(p-1)^{2}}$ ${\displaystyle p}$
• Root criterion : convergence criterion for series

Y

• Yoneda -Lemma : Statement about the set of natural transformations between a Hom functor and another functor.
• Young's theorem : The set of discontinuities in a function is a set. ${\displaystyle F_{\sigma }}$

Z

• Tameness theorem : Complete, 3-dimensional, hyperbolic manifolds with finitely generated fundamental groups are topologically tame.
• Lemma of Zabreiko : A statement of continuity about certain subadditive functionals on Banach spaces
• Zassenhaus Lemma : A Technical Isomorphism Theorem for Groups (Butterfly Lemma )
• Central limit theorem : For every sequence of stochastically independent, identically distributed real random variables for which the expected value and variance exist, the sequence of the distributions of the standardized sum variables converges weakly to the standard normal distribution.
• Multi-dimensional central limit theorem : convergence in distribution of rescaled sums of random vectors against the multi-dimensional normal distribution
• Lemma von Zolotareff : The Legendre symbol is equal to the sign of a special permutation.
• Zorn's Lemma : Every non-empty semi-ordered set in which every chain (i.e. every totally ordered subset) has an upper bound contains at least one maximal element.
• Two-squares theorem : Every prime number of the form 4 n +1 can be written as the sum of two square numbers.
• Second isomorphism theorem : If we are normal divisors, then applies ${\displaystyle N\subset H\subset G}$ ${\displaystyle (G/N)/(H/N)\cong G/H.}$
• Intermediate value theorem : A continuous function takes between and all values ​​between and . ${\displaystyle f}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle f(a)}$ ${\displaystyle f(b)}$

See also

• List of eminent mathematicians