** Artin reciprocity law, a general theorem in number theory that provided a partial solution to Hilbert's ninth problem

** Every vector space has a basis.

** Every field extension has a transcendence basis.

** The theorem that every Hilbert space has an orthonormal basis.

** Dual basis

** Orthonormal basis

** Schauder basis

** Gröbner basis

** Amongst them, there are those who hold that moral knowledge is gained inferentially on the basis of some sort of non-moral epistemic process, as opposed to ethical intuitionism.

** the Almagest which becomes the basis for western and Middle Eastern astronomy until the time of Copernicus and Kepler ;

** Oral arguments begin in the landmark U. S. Supreme Court case Loving v. Virginia, 388 U. S. 1 ( 1967 ), challenging the State of Virginia's statutory scheme to prevent marriages between persons solely on the basis of racial classifications.

** In response to the events of March 7 and 9 in Selma, Alabama, President Lyndon B. Johnson sends a bill to Congress that forms the basis for the Voting Rights Act of 1965.

** Conon of Samos, Greek mathematician and astronomer whose work on conic sections ( curves of the intersections of a right circular cone with a plane ) serves as the basis for the fourth book of the Conics of Apollonius of Perga ( b. c. 280 BC )

** an operation is performed on the basis of an instruction

** " Instrumentalist perennialism ", while seeing ethnicity primarily as a versatile tool that identified different ethnics groups and limits through time, explains ethnicity as a mechanism of social stratification, meaning that ethnicity is the basis for a hierarchical arrangement of individuals.

** Linear algebra explains how a vector space can be given a basis, and then any vector can be expressed in this basis.

** Die Seeraeuber ( The Pirate ), a play by Ludwig Fulda and the basis of the Behrman play

** Hui Lan Ji ( 灰闌記 ， 灰兰记 ), by Li Xingdao ( 李行道 ) became the basis for The Caucasian Chalk Circle

** Jeddah for Saudia on wet-lease basis

** Illustration the original imagery that was used as a basis for illustration

** a type of flower head where the bracts are located under the basis, such as a daisy's

** 320 / 640 × 200, horizontal resolution selectable on a line by line basis

** This level of clearance will grant the right to access designated documents with markings of Protected A, B & C information / assets on a need-to-know basis.

** This level of clearance will grant the right to access designated and classified information up to Confidential level on a need-to-know basis.

** Well-ordering theorem: Every set can be well-ordered.

** Tarski's theorem: For every infinite set A, there is a bijective map between the sets A and A × A.

** König's theorem: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger cardinals.

** Tychonoff's theorem stating that every product of compact topological spaces is compact.

** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.

** The Vitali theorem on the existence of non-measurable sets which states that there is a subset of the real numbers that is not Lebesgue measurable.

** Stone's representation theorem for Boolean algebras needs the Boolean prime ideal theorem.

** The Nielsen – Schreier theorem, that every subgroup of a free group is free.

** The Hahn – Banach theorem in functional analysis, allowing the extension of linear functionals

** The Banach – Alaoglu theorem about compactness of sets of functionals.

** The Baire category theorem about complete metric spaces, and its consequences, such as the open mapping theorem and the closed graph theorem.

** Gödel's completeness theorem for first-order logic: every consistent set of first-order sentences has a completion.

** The numbers and are not algebraic numbers ( see the Lindemann – Weierstrass theorem ); hence they are transcendental.

** Bayes ' theorem

** More generally, Rademacher's theorem extends the differentiability result to Lipschitz mappings between Euclidean spaces: a Lipschitz map ƒ: U → R < sup > m </ sup >, where U is an open set in R < sup > n </ sup >, is almost everywhere differentiable.

** Lyapunov's central limit theorem

** Superposition theorem, in electronics

** " Kelvin's vorticity theorem for incompressible or barotropic flow ".

** Various proofs of the four colour theorem.