Representation theorem

In mathematics, a representation theorem is a theorem that states that every abstract structure with certain properties is isomorphic to a 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.

• in category theory,
  • the Yoneda lemma explains how arbitrary functors into the category of sets can be seen as hom functors
• in set theory,
  • Mostowski's collapsing theorem states that every well-founded extensional structure is isomorphic to a transitive set with the ∈-relation
• in functional analysis
  • 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.