# 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.[1]

## Examples

### Algebra

- Cayley's theorem states that every group is isomorphic to a subgroup of a permutation group.[2]
- Representation theory studies properties of abstract groups via their representations as linear transformations of vector spaces.[1]
- Stone's representation theorem for boolean algebras states that every Boolean algebra is isomorphic to a field of sets.[3]
- 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*.[4] - 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.[5]
- 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 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*.

### 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.[6]

## References

- "The Definitive Glossary of Higher Mathematical Jargon".
*Math Vault*. 2019-08-01. Retrieved 2019-12-08. - "Cayley's Theorem and its Proof".
*www.sjsu.edu*. Retrieved 2019-12-08. - Dirks, Matthew. "The Stone Representation Theorem for Boolean Algebras" (PDF).
*math.uchicago.edu*. Retrieved 2019-12-08. - Schneider, Friedrich Martin (November 2017). "A uniform Birkhoff theorem".
*Algebra universalis*.**78**(3): 337–354. doi:10.1007/s00012-017-0460-1. ISSN 0002-5240. - "Freyd-Mitchell embedding theorem in nLab".
*ncatlab.org*. Retrieved 2019-12-08. - "Notes on the Nash embedding theorem".
*What's new*. 2016-05-11. Retrieved 2019-12-08.

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