Representation theorem

In mathematics, a representation theorem is a theorem that states that every abstract structure with certain properties is isomorphic to another (abstract or concrete) structure.

Examples

Algebra

Cayley's theorem states that every group is isomorphic to a permutation group.
Representation theory studies properties of abstract groups via their representations as linear transformations of vector spaces.
Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a field of sets.
A variant, Stone's representation theorem for distributive lattices, states that every distributive lattice is isomorphic to a sublattice of the power set lattice of some set.

Another variant, Stone's duality, states that there exists a duality (in the sense of an arrow-reversing equivalence) between the categories of Boolean algebras and that of Stone spaces.
The Poincaré–Birkhoff–Witt theorem states that every Lie algebra embeds into the commutator Lie algebra of its universal enveloping algebra.
Ado's theorem states that every finite-dimensional Lie algebra over a field of characteristic zero embeds into the Lie algebra of endomorphisms of some finite-dimensional vector space.
Birkhoff's HSP theorem states that every model of an algebra A is the homomorphic image of a subalgebra of a direct product of copies of A.
In the study of semigroups, the Wagner–Preston theorem provides a representation of an inverse semigroup S, as a homomorphic image of the set of partial bijections on S, and the semigroup operation given by composition.

Category theory

The Yoneda lemma provides a full and faithful limit-preserving embedding of any category into a category of presheaves.
Mitchell's embedding theorem for abelian categories realises every small abelian category as a full (and exactly embedded) subcategory of a category of modules over some ring.
Mostowski's collapsing theorem states that every well-founded extensional structure is isomorphic to a transitive set with the ∈-relation.
One of the fundamental theorems in sheaf theory states that every sheaf over a topological space can be thought of as a sheaf of sections of some (étalé) bundle over that space: the categories of sheaves on a topological space and that of étalé spaces over it are equivalent, where the equivalence is given by the functor that sends a bundle to its sheaf of (local) sections.

Functional analysis

The Gelfand–Naimark–Segal construction embeds any C*-algebra in an algebra of bounded operators on some Hilbert space.
The Gelfand representation (also known as the commutative Gelfand–Naimark theorem) states that any commutative C*-algebra is isomorphic to an algebra of continuous functions on its Gelfand spectrum. It can also be seen as the construction as a duality between the category of commutative C*-algebras and that of compact Hausdorff spaces.
The Riesz representation theorem states that a Hilbert space, such as the square-integrable function space L²(X) on a manifold X, any linear functional F is equal to the inner product with a fixed element $a\in H$ , i.e. $F(v)=\langle a,v\rangle$ for all $v\in H$ . The more general Riesz–Markov–Kakutani representation theorem has several versions, one of them identifying the dual space of C₀(X) with the set of regular measures on X.

Geometry

The Whitney embedding theorems embed any abstract manifold in some Euclidean space.
The Nash embedding theorem embeds an abstract Riemannian manifold isometrically in a Euclidean space.

Order Theory

It is a basic result that every partially ordered set is isomorphic to a collection of sets ordered by inclusion (containment).

Economics

A preference representation theorem states conditions for the existence of a utility function representing a preference relation. Examples are Von Neumann–Morgenstern utility theorem and Debreu's representation theorems.