Representation theorem

Short description: Proof that every structure with certain properties is isomorphic to another structure

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.

## Contents

* 1 Examples
* 1.1 Algebra
* 1.2 Category theory
* 1.3 Functional analysis
* 1.4 Geometry
* 1.5 Order Theory
* 1.6 Economics
* 2 See also
* 3 References

## Examples

### Algebra

* Cayley's theorem states that every group is isomorphic to a permutation group.[1]
* 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.[2]

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.[3]
* 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.[4]
* 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∈H, i.e. F(v)=⟨a,v⟩ for all v∈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.[5]

### 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.

## See also

* Classification theorem - Describes the objects of a given type, up to some equivalence

## References

1. ↑ "Cayley's Theorem and its Proof". http://www.sjsu.edu/faculty/watkins/cayleyth.htm.
2. ↑ Dirks, Matthew. "The Stone Representation Theorem for Boolean Algebras". http://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Dirks.pdf.
3. ↑ 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.
4. ↑ Freyd–Mitchell embedding theorem in nLab
5. ↑ "Notes on the Nash embedding theorem" (in en). 2016-05-11. https://terrytao.wordpress.com/2016/05/11/notes-on-the-nash-embedding-theorem/.

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