Representation theorem - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, 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
  Cayley's theorem
  In group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group acting on G...
  
  states that every group
  Group (mathematics)
  In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
  
  is isomorphic to a transformation group on some set.
  
  Representation theory
  Representation theory
  Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
  
  studies properties of abstract groups via their representations as linear transformations of vector spaces.
- Stone's representation theorem for Boolean algebra
  Boolean algebra
  In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets...
  
  s 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
  Distributive lattice
  In mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection...
  
  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 algebra
  Boolean algebra
  In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets...
  
  s and that of Stone spaces.
- The Poincaré-Birkhoff-Witt theorem states that every Lie algebra
  Lie algebra
  In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
  
  embeds into the commutator Lie algebra of its universal enveloping algebra
  Universal enveloping algebra
  In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie algebra...
  
  .
- Ado's theorem
  Ado's theorem
  In abstract algebra, Ado's theorem states that every finite-dimensional Lie algebra L over a field K of characteristic zero can be viewed as a Lie algebra of square matrices under the commutator bracket...
  
  states that every finite dimensional Lie algebra
  Lie algebra
  In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
  
  over a field
  Field
  -Places:* Field, British Columbia, Canada* Field, Minneapolis, Minnesota, United States* Field, Ontario, Canada* Field Island, Nunavut, Canada* Mount Field - Expanses of open ground :* Field...
  
  of characteristic zero embeds into the Lie algebra of endomorphisms of some finite dimensional vector space.
in category theory,
- The Yoneda lemma
  Yoneda lemma
  In mathematics, specifically in category theory, the Yoneda lemma is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory...
  
  provides an full and faithful limit preserving embedding of any category into a category of presheaves.
- Mitchell's embedding theorem
  Mitchell's embedding theorem
  Mitchell's embedding theorem, also known as the Freyd–Mitchell theorem, is a result stating that every abelian category admits a full and exact embedding into the category of R-modules...
  
  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
  Sheaf
  A sheaf is one of the large bundles in which cereal plants are bound after reaping. Accounts of two usages derived from this are found at:* Sheaf * Sheaf toss -Other:...
  
  theory states that every sheaf over a topological space
  Topological space
  Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
  
  can be thought of as a sheaf of section
  Section
  Section may refer to:* Section * Section * Archaeological section* Histological section, a thin slice of tissue used for microscopic examination* Section, an instrumental group within an orchestra...
  
  s of some (Étale) bundle over that space: the categories of sheaves on a topological space and that of Étale 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
  Hilbert space
  The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
  
  .
- The Gelfand representation
  Gelfand representation
  In mathematics, the Gelfand representation in functional analysis has two related meanings:* a way of representing commutative Banach algebras as algebras of continuous functions;...
  
  (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
  Riesz representation theorem
  There are several well-known theorems in functional analysis known as the Riesz representation theorem. They are named in honour of Frigyes Riesz.- The Hilbert space 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
  Manifold
  In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
  
  in some Euclidean space
  Euclidean space
  In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
  
  .
- The Nash embedding theorem
  Nash embedding theorem
  The Nash embedding theorems , named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path...
  
  embeds an abstract Riemannian manifold
  Riemannian manifold
  In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
  
  isometrically in an Euclidean space
  Euclidean space
  In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
  
  .

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