Representation theorem
From Wikipedia, the free encyclopedia
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.
- Cayley's theorem states that every group is isomorphic to a transformation group on 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.