List of lemmas

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This following is a list of lemmas (or, "lemmata", i.e. minor theorems, or sometimes intermediate technical results factored out of proofs). See also list of axioms, list of theorems and list of conjectures.

Algebra

Main category: Lemmas in algebra

• Abhyankar's lemma
• Aubin–Lions lemma
• Bergman's diamond lemma
• Fitting lemma
• Injective test lemma
• Hua's lemma (exponential sums)
• Krull's separation lemma
• Schanuel's lemma (projective modules)
• Schwartz–Zippel lemma
• Shapiro's lemma
• Stewart–Walker lemma (tensors)
• Whitehead's lemma (Lie algebras)
• Zariski's lemma

Algebraic geometry

• Abhyankar's lemma
• Fundamental lemma (Langlands program)

Category theory

A commutative diagram that illustrates the five lemma

• Five lemma
• Horseshoe lemma
• Nine lemma
• Short five lemma
• Snake lemma
• Splitting lemma

Linear algebra

• Matrix determinant lemma
• Matrix inversion lemma

Group theory

• Burnside's lemma also known as the Cauchy–Frobenius lemma
• Frattini's lemma (finite groups)
• Goursat's lemma
• Mautner's lemma (representation theory)
• Ping-pong lemma (geometric group theory)
• Schreier's subgroup lemma
• Schur's lemma (representation theory)
• Zassenhaus lemma

Polynomials

• Gauss's lemma (polynomials)
• Schwartz–Zippel lemma

Ring theory and commutative algebra

• Artin–Rees lemma
• Hensel's lemma (commutative rings)
• Nakayama lemma
• Noether's normalization lemma
• Prime avoidance lemma

Universal algebra

• Jónsson's lemma

Analysis

Main category: Lemmas in analysis

• Fekete's lemma
• Fundamental lemma of calculus of variations
• Hopf lemma
• Sard's lemma (singularity theory)
• Stechkin's lemma (functional and numerical analysis)
• Vitali covering lemma (real analysis)
• Watson's lemma

Complex analysis

• Estimation lemma (contour integrals)
• Hartogs's lemma (several complex variables)
• Jordan's lemma
• Lemma on the Logarithmic derivative
• Schwarz lemma

Fourier analysis

• Riemann–Lebesgue lemma

Differential equations

• Borel's lemma (partial differential equations)
• Grönwall's lemma
• Lax–Milgram lemma
• Pugh's closing lemma
• Weyl's lemma (Laplace equation) (partial differential equations)

Differential forms

• Poincaré lemma of closed and exact differential forms

Functional analysis

• Cotlar–Stein lemma
• Ehrling's lemma
• Riesz's lemma

Mathematical series

• Abel's lemma
• Kronecker's lemma

Numerical analysis

• Bramble–Hilbert lemma
• Céa's lemma

Applied mathematics

diagram of a length-8 fast Fourier transform

• Danielson–Lanczos lemma (Fourier transforms)
• Farkas's lemma (linear programming)
• Feld–Tai lemma (electromagnetism)
• Little's lemma (queuing theory)
• Finsler's lemma

Control theory

• Finsler's lemma
• Hautus lemma
• Kalman–Yakubovich–Popov lemma

Computational complexity theory

• Isolation lemma
• Switching lemma

Cryptography

• Forking lemma
• Leftover hash lemma
• Piling-up lemma (linear cryptanalysis)
• Yao's XOR lemma

Formal languages

• Interchange lemma
• Newman's lemma (term rewriting)
• Ogden's lemma
• Pumping lemma sometimes called the Bar-Hillel lemma

Microeconomics

• Hotelling's lemma
• Shephard's lemma

Combinatorics

• Cousin's lemma (integrals)
• Dickson's lemma
• Littlewood–Offord lemma
• Pólya–Burnside lemma
• Sperner's lemma
• Ky Fan lemma (combinatorial geometry)

Graph theory

Main category: Lemmas in graph theory

• Berge's lemma
• Counting lemma
• Crossing lemma
• Expander mixing lemma
• Handshaking lemma
• Kelly's lemma
• Kőnig's lemma
• Szemerédi regularity lemma

Order theory

• Higman's lemma
• Ultrafilter lemma

Dynamical systems

• Barbalat's lemma
• Kac's lemma (ergodic theory)

Geometry

• Shadowing lemma
• Big-little-big lemma (mathematics of paper folding)
• Gordan's lemma
• Hilbert's lemma

Euclidean geometry

• Archimedes's lemmas
• Johnson–Lindenstrauss lemma (Euclidean geometry)

Hyperbolic geometry

• Margulis lemma

Metric spaces

• Lebesgue's number lemma (dimension theory)

Riemannian geometry

• Gauss's lemma (Riemannian geometry)

Mathematical logic

• Craig interpolation lemma
• Diagonal lemma
• Lindenbaum's lemma
• Mostowski collapse lemma
• Teichmüller–Tukey lemma also known as Tukey's lemma
• Zorn's lemma; equivalent to the axiom of choice

Set theory

Main category: Lemmas in set theory

• Covering lemma
• Delta lemma
• Dynkin lemma
• Fodor's lemma
• Fixed-point lemma for normal functions (axiomatic set theory)
• Moschovakis coding lemma
• Rasiowa–Sikorski lemma

Number theory

Main category: Lemmas in number theory

• Bézout's lemma
• Dwork's lemma
• Euclid's lemma
• Gauss's lemma
• Hensel's lemma
• Zolotarev's lemma
• Siegel's lemma (Diophantine approximation)

Analytic number theory

• Hua's lemma
• Vaughan's lemma

Diophantine equations

• Bhaskara's lemma

Sieve theory

• Fundamental lemma of sieve theory

Probability theory

• Borel–Cantelli lemma
• Doob–Dynkin lemma
• Itô's lemma (stochastic calculus)
• Lovász local lemma
• Stein's lemma
• Wald's lemma

Statistics

• Glivenko–Cantelli lemma
• Neyman–Pearson lemma
• Robbins lemma

Measure theory

• Factorization lemma
• Fatou's lemma
• Frostman's lemma (geometric measure theory)
• Malliavin's absolute continuity lemma

Topology

• Lindelöf's lemma
• Urysohn's lemma
• Tube lemma

Differential topology

• Morse lemma

Fixed-point theory

An example of a covering described by the Knaster–Kuratowski–Mazurkiewicz lemma

• Knaster–Kuratowski–Mazurkiewicz lemma

Geometric topology

• Dehn's lemma

Topological groups and semigroups

• Ellis–Numakura lemma (topological semigroups)