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.
• 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 C₀(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.