List of theorems - Definition and Overview

This is a list of theorems, by Wikipedia page. See also

• list of fundamental theorems
• list of lemmas
• list of conjectures
• list of mathematical proofs

In some fields, theorem can be considered as a courtesy title, given to major results, although with a content that would not satisfy a mathematician. No attempt is made here to comment on that aspect of usage: this is a list of results known as theorems. Most of the results do come from mathematics, but there are others from theoretical physics, economics and so on.

A

• Abel's theorem (mathematical analysis)
• Abelian and tauberian theorems (mathematical analysis)
• Abel-Ruffini theorem (theory of equations, Galois theory)
• Ankeny-Artin-Chowla theorem (number theory)
• Arrow's impossibility theorem (game theory)
• Artin-Schreier theorem (real closed fields)
• Artin-Wedderburn theorem (abstract algebra)
• Arzelà-Ascoli theorem (functional analysis)
• Atiyah-Singer index theorem (elliptic differential operators, harmonic analysis)

B

• Baire category theorem (topology, metric spaces)
• Banach-Alaoglu theorem (functional analysis)
• Banach fixed point theorem (metric spaces, differential equations)
• Banach-Steinhaus theorem (functional analysis)
• Barbier's theorem (geometry)
• Bass's theorem (group theory)
• Bayes' theorem (probability)
• Beatty's theorem (diophantine approximation)
• Beck's theorem (category theory, or geometry)
• Bell's theorem (quantum theory - physics)
• Bendixson-Dulac theorem (dynamical systems)
• Berry-Esséen theorem (probability theory)
• Bertrand's ballot theorem (probability theory, combinatorics)
• Bertrand's postulate (prime numbers)
• Bezout's theorem (algebraic curves)
• Binomial theorem (algebra, combinatorics)
• Birkhoff's theorem (ergodic theory)
• Bohr-Mollerup theorem (gamma function)
• Bolyai-Gerwien theorem (geometry)
• Bolzano-Weierstrass theorem (real analysis, calculus)
• Boolean prime ideal theorem (mathematical logic)
• Borel-Bott-Weil theorem (representation theory)
• Bott periodicity theorem (homotopy theory)
• Borsuk-Ulam theorem (topology)
• Brouwer fixed point theorem (topology)
• Brown's representability theorem (homotopy theory)
• Bruck-Chowla-Ryser theorem (combinatorics)
• Buckingham Pi theorem (dimensional analysis)

C

• Cantor-Bernstein-Schroeder theorem (Set theory, cardinal numbers)
• Cantor's theorem (Set theory, Cantor's diagonal argument)
• Carathéodory's theorem (conformal mapping)
• Carathéodory's theorem (convex hull)
• Carathéodory's theorem (measure theory)
• Cartan's theorem (Lie group)
• Cartan's theorems A and B (several complex variables)
• Cauchy integral theorem (Complex analysis)
• Cayley-Hamilton theorem (Linear algebra)
• Cayley's theorem (group theory)
• Central limit theorem (probability)
• Ceva's theorem (geometry)
• Chebotarev's density theorem (number theory)
• Chern-Gauss-Bonnet theorem (differential geometry)
• Chinese remainder theorem (number theory)
• Chowla-Mordell theorem (number theory)
• Church-Rosser theorem (lambda calculus)
• Closed graph theorem (functional analysis)
• Cluster decomposition theorem (quantum field theory)
• Coase theorem (economics)
• Cochran's theorem (statistics)
• Compactness theorem (mathematical logic)
• Convolution theorem (Fourier transforms)
• Cook's theorem (computational complexity theory)
• Cox's theorem (probability foundations)
• Cut-elimination theorem (proof theory)

D

• Dandelin's theorem (geometry)
• De Branges' theorem (complex analysis)
• De Finetti's theorem (probability)
• De Rham's theorem (differential topology)
• Deduction theorem (logic)
• Desargues' theorem (geometry)
• Descartes' theorem (geometry)
• Dilworth's theorem (combinatorics, order theory)
• Dimension theorem for vector spaces (vector spaces, linear algebra)
• Dirichlet's theorem on arithmetic progressions (number theory)
• Dirichlet's unit theorem (algebraic number theory)
• Divergence theorem (vector calculus)
• Dominated convergence theorem (Lebesgue integration)

E

• Earnshaw's theorem (electrostatics)
• Ehresmann's theorem (differential topology)
• Equipartition theorem (ergodic theory)
• Erdős-Ko-Rado theorem (combinatorics)
• Euler's rotation theorem (geometry)
• Euler's theorem (number theory)
• Euler's theorem on homogeneous functions (multivariate calculus)
• Extreme value theorem

F

• Faltings' theorem (diophantine geometry)
• Fermat's last theorem (number theory)
• Fermat's little theorem (number theory)
• Fisher separation theorem (economics)
• Fixed point theorems in infinite-dimensional spaces
• Fluctuation dissipation theorem (physics)
• Fluctuation theorem
• Four color theorem (graph theory)
• Fourier inversion theorem (harmonic analysis)
• Frobenius reciprocity theorem (group representations)
• Frobenius theorem (foliations)
• Fubini's theorem (integration)
• Fuglede's theorem (functional analysis)
• Fundamental theorem of algebra (complex analysis)
• Fundamental theorem of arbitrage-free pricing (financial mathematics)
• Fundamental theorem of arithmetic (number theory)
• Fundamental theorem of calculus (calculus)
• Fundamental theorem of poker (poker)
• Fundamental theorem on homomorphisms (abstract algebra)

G

• Gauss theorem (vector calculus)
• Gauss's Theorema Egregium (differential geometry)
• Gauss-Bonnet theorem (differential geometry)
• Gauss-Markov theorem (statistics)
• Gauss-Wantzel theorem (geometry)
• Gelfand-Naimark theorem (functional analysis)
• Gelfond-Schneider theorem (transcendence theory)
• Gibbard-Satterthwaite theorem (voting methods)
• Girsanov's theorem (stochastic processes)
• Goddard-Thorn theorem (vertex algebras)
• Gödel's completeness theorem (mathematical logic)
• Gödel's incompleteness theorem (mathematical logic)
• Goodstein's theorem (mathematical logic)
• Green's theorem (vector calculus)
• Gromov's compactness theorem (Riemannian geometry)
• Gromov's theorem (group theory)
• Gromov-Ruh theorem (differential geometry)

H

• H-theorem (thermodynamics)
• Haag's theorem (quantum field theory)
• Haboush's theorem (algebraic groups, representation theory, invariant theory)
• Hadwiger's theorem (geometry, measure theory)
• Hahn embedding theorem (ordered groups)
• Hairy ball theorem (algebraic topology)
• Hahn-Banach theorem (functional analysis)
• Hales-Jewett theorem (combinatorics)
• Ham sandwich theorem (topology)
• Heine-Borel theorem (real analysis)
• Hellinger-Toeplitz theorem (functional analysis)
• Helly's theorem (convex sets)
• Herbrand-Ribet theorem (cyclotomic fields)
• Hilbert's basis theorem (commutative algebra,invariant theory)
• Hilbert's Nullstellensatz (theorem of zeroes) (commutative algebra, algebraic geometry)
• Hilbert-Speiser theorem (cyclotomic fields)
• Hopf-Rinow theorem (differential geometry)
• Hurewicz theorem (algebraic topology)
• Hurwitz's automorphisms theorem (algebraic curves)

I

• Intermediate value theorem (calculus)
• Implicit function theorem (vector calculus)
• Infinite monkey theorem (probability)
• Inverse function theorem (vector calculus)
• Isomorphism theorem (abstract algebra)
• Isoperimetric theorem (curves, calculus of variations)

J

• Jacobson density theorem (ring theory)
• Jordan curve theorem (topology)
• Jordan-Hölder theorem (group theory)
• Jordan-Schönflies theorem (geometric topology)

K

• Kirchoff's theorem
• Kirszbraun theorem (Lipschitz continuity)
• Kleene's recursion theorem (recursion theory)
• Knaster-Tarski theorem (order theory)
• Kolmogorov-Arnold-Moser theorem (dynamical systems)
• König's theorem (mathematical logic)
• Kronecker's theorem (diophantine approximation)
• Kronecker-Weber theorem (number theory)
• Krull's principal ideal theorem (commutative algebra)
• Künneth theorem (algebraic topology)

L

• Lagrange's theorem (group theory)
• Lagrange's four-square theorem (number theory)
• Lagrange inversion theorem (mathematical analysis, combinatorics)
• Lagrange reversion theorem (mathematical analysis, combinatorics)
• Lami's theorem (statics)
• Laurent expansion theorem (complex analysis)
• Lefschetz fixed point theorem (algebraic topology)
• Lehmann-Scheffé theorem (statistics)
• Lindemann-Weierstrass theorem (transcendence theory)
• Lie-Kolchin theorem (algebraic groups, representation theory)
• Linear congruence theorem (number theory, modular arithmetic)
• Linear speedup theorem (computational complexity theory)
• Linnik's theorem (number theory)
• Liouville's theorem (complex analysis) (entire functions)
• Liouville's theorem (Hamiltonian) (Hamiltonian mechanics)
• Löb's theorem (mathematical logic)
• Löwenheim-Skolem theorem (mathematical logic)
• Lyapunov's central limit theorem (probability theory)

M

• Mahler's compactness theorem (geometry of numbers)
• Mahler's theorem (p-adic analysis)
• Marcinkiewicz theorem (functional analysis)
• Marriage theorem (combinatorics)
• Master theorem (recurrence relations, asymptotic analysis)
• Maschke's theorem (group representations)
• Matiyasevich's theorem (mathematical logic)
• Max flow min cut theorem (graph theory)
• Maximum power theorem (electrical circuits)
• Maxwell's theorem (probability theory)
• Mean value theorem (calculus)
• Mercer's theorem (functional analysis)
• Mertens' theorems (number theory)
• Metrization theorems (topological spaces)
• Min-max theorem (functional analysis)
• Minimax theorem
• Minkowski's theorem (geometry of numbers)
• Mitchell's embedding theorem (category theory)
• Mohr-Mascheroni theorem (geometry)
• Monotone convergence theorem (mathematical analysis)
• Mordell-Weil theorem (number theory)
• Morera's theorem (complex analysis)
• Morley's categoricity theorem (model theory)
• Morley's trisector theorem (geometry)
• Multinomial theorem (algebra, combinatorics)
• Myers theorem (differential geometry)
• Myhill-Nerode theorem (formal languages)

N

• Nagell-Lutz theorem (elliptic curves)
• Nash embedding theorem (differential geometry)
• Nielsen-Schreier theorem (free groups)
• No cloning theorem (quantum computation)
• Noether's theorem (Lie groups, calculus of variations, differential invariants, physics)
• No-ghost theorem (vertex algebras)
• Norton's theorem (electrical networks)
• Nyquist-Shannon sampling theorem (information theory)

O

• Open mapping theorem (functional analysis)

P

• Paley-Wiener theorem (Fourier transforms)
• Pappus's centroid theorem (geometry)
• Parseval's theorem (Fourier analysis)
• Pascal's theorem (conics)
• Pentagonal number theorem (number theory)
• Perfect graph theorem (graph theory)
• Peter-Weyl theorem (representation theory)
• Picard theorem (complex analysis)
• Picard-Lindelöf theorem (ordinary differential equations)
• Pick's theorem (geometry)
• Pitman-Koopman-Darmois theorem (statistics)
• Plancherel theorem (Fourier analysis)
• Poincaré-Birkhoff-Witt theorem (universal enveloping algebras)
• Poincaré duality theorem (algebraic topology of manifolds)
• Poncelet-Steiner theorem (geometry)
• Post's theorem (mathematical logic)
• Prime number theorem (number theory)
• Primitive element theorem (field theory)
• Pythagorean theorem (geometry)

R

• Radon's theorem (convex sets)
• Radon-Nikodym theorem (measure theory)
• Ramsey's theorem (graph theory,combinatorics)
• Rank-nullity theorem (linear algebra)
• Rao-Blackwell theorem (statistics)
• Rational root theorem (algebra,polynomials)
• Reeh-Schlieder theorem (local quantum field theory)
• Residue theorem (complex analysis)
• Rice's theorem (recursion theory, computer science)
• Riemann mapping theorem (complex analysis)
• Riemann-Roch theorem (Riemann surfaces, algebraic curves)
• Riesz representation theorem (functional analysis,Hilbert space)
• Riesz-Thorin theorem (functional analysis)
• Robertson-Seymour theorem (graph theory)
• Rolle's theorem (calculus)
• Roth's theorem (diophantine approximation)
• Rouché's theorem (complex analysis)

S

• Sahlqvist correspondence theorem (modal logic)
• Sarkovskii's theorem (dynamical systems)
• Savitch's theorem (computational complexity theory)
• Schauder fixed point theorem (functional analysis)
• Schreier refinement theorem (group theory)
• Seifert-van Kampen theorem (algebraic topology)
• Shannon's theorem (information theory)
• Simplicial approximation theorem (algebraic topology)
• Skolem-Noether theorem (simple algebras)
• Soundness theorem (mathematical logic)
• Space hierarchy theorem (computational complexity theory)
• Spectral theorem (functional analysis)
• Speedup theorem (computational complexity theory)
• Sperner's theorem (combinatorics)
• Spin-statistics theorem (physics)
• Sprague-Grundy theorem (combinatorial game theory)
• Squeeze theorem (mathematical analysis)
• Stanley's reciprocity theorem (combinatorics)
• Stark-Heegner theorem (number theory)
• Stokes' theorem (vector calculus, differential topology)
• Stolper-Samuelson theorem (economics)
• Stone's representation theorem for Boolean algebras (mathematical logic)
• Stone's theorem on one-parameter unitary groups (functional analysis)
• Stone-Tukey theorem (topology)
• Stone-von Neumann theorem (functional analysis, representation theory of the Heisenberg group, quantum mechanics)
• Stone-Weierstrass theorem (functional analysis)
• Swan's theorem (module theory)
• Sylow theorem (group theory)
• Sylvester's theorem (number theory)
• Sylvester-Gallai theorem (plane geometry)
• Szemerédi's theorem (combinatorics)
• Szemerédi-Trotter theorem (combinatorics)

T

• Takagi existence theorem (number theory)
• Taniyama-Shimura theorem (number theory)
• Tarski's indefinability theorem (mathematical logic)
• Taylor's theorem (calculus)
• Thales' theorem (geometry)
• Thevenin's theorem (electrical circuits)
• Thue-Siegel-Roth theorem (diophantine approximation)
• Tietze extension theorem (general topology)
• Tikhonov fixed point theorem (functional analysis)
• Time hierarchy theorem (computational complexity theory)
• Turán's theorem (graph theory)
• Tychonoff's theorem (general topology)

U

• Uniformization theorem (complex analysis, differential geometry)
• Universal coefficient theorem (algebraic topology)

V

• Van der Waerden's theorem (combinatorics)
• Virial theorem (classical mechanics)
• Vitali theorem (measure theory)
• Vitali-Hahn-Saks theorem (measure theory)
• Von Neumann bicommutant theorem (functional analysis)

W

• Weierstrass-Casorati theorem (complex analysis)
• Weierstrass preparation theorem (several complex variables,commutative algebra)
• Well-ordering theorem (mathematical logic)
• Whitehead theorem (homotopy theory)
• Whitney embedding theorem (differential manifolds)
• Wigner-Eckhart theorem (Clebsch-Gordan coefficients)
• Wilson's theorem (number theory)
• Wolstenholme's theorem (number theory)

Z

• Zeckendorf's theorem
• Zermelo's theorem[1] (http://www.math.harvard.edu/~elkies/FS23j.03/zermelo.pdf) (game theory)