# Representation theorem

See also: Universal approximation 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.

For example,

- in algebra,
- Cayley's theorem states that every group is isomorphic to a transformation group on some set.
- 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 lattices states that every distributive lattice is isomorphic to a sublattice of the power set lattice of some set.
- Another variant 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*.

- Cayley's theorem states that every group is isomorphic to a transformation group on some set.
- in 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.

- in 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 is actually a list of several theorems; one of them identifies the dual space of
*C*_{0}(*X*) with the set of regular measures on*X*.

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

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