Representation theorem

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 "C"0("X") with the set of regular measures on "X".

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