List of lemmas

This is a dynamic list and may never be able to satisfy particular standards for completeness. You can help by adding missing items with reliable sources.

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.

This list is incomplete; you can help by adding missing items. (August 2008)

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## Algebra[edit]

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[edit]

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

### Category theory[edit]

A commutative diagram that illustrates the five lemma

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

### Linear algebra[edit]

* Matrix determinant lemma
* Matrix inversion lemma

### Group theory[edit]

* 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[edit]

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

### Ring theory and commutative algebra[edit]

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

### Universal algebra[edit]

* Jónsson's lemma

## Analysis[edit]

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[edit]

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

### Fourier analysis[edit]

* Riemann–Lebesgue lemma

### Differential equations[edit]

* 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[edit]

* Poincaré lemma of closed and exact differential forms

### Functional analysis[edit]

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

### Mathematical series[edit]

* Abel's lemma
* Kronecker's lemma

### Numerical analysis[edit]

* Bramble–Hilbert lemma
* Céa's lemma

## Applied mathematics[edit]

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[edit]

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

### Computational complexity theory[edit]

* Isolation lemma
* Switching lemma

#### Cryptography[edit]

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

#### Formal languages[edit]

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

### Microeconomics[edit]

* Hotelling's lemma
* Shephard's lemma

## Combinatorics[edit]

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

### Graph theory[edit]

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[edit]

* Higman's lemma
* Ultrafilter lemma

## Dynamical systems[edit]

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

## Geometry[edit]

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

### Euclidean geometry[edit]

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

### Hyperbolic geometry[edit]

* Margulis lemma

### Metric spaces[edit]

* Lebesgue's number lemma (dimension theory)

### Riemannian geometry[edit]

* Gauss's lemma (Riemannian geometry)

## Mathematical logic[edit]

* 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[edit]

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[edit]

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[edit]

* Hua's lemma
* Vaughan's lemma

### Diophantine equations[edit]

* Bhaskara's lemma

### Sieve theory[edit]

* Fundamental lemma of sieve theory

## Probability theory[edit]

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

### Statistics[edit]

* Glivenko–Cantelli lemma
* Neyman–Pearson lemma
* Robbins lemma

### Measure theory[edit]

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

## Topology[edit]

* Lindelöf's lemma
* Urysohn's lemma
* Tube lemma

### Differential topology[edit]

* Morse lemma

### Fixed-point theory[edit]

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

* Knaster–Kuratowski–Mazurkiewicz lemma

### Geometric topology[edit]

* Dehn's lemma

### Topological groups and semigroups[edit]

* Ellis–Numakura lemma (topological semigroups)