Representation theorem

• Cayley's theorem states that every group is isomorphic to a permutation group.{{Cite web|url=http://www.sjsu.edu/faculty/watkins/cayleyth.htm|title=Cayley's Theorem and its Proof|website=www.sjsu.edu|access-date=2019-12-08}}
• 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.{{Cite web|url=http://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Dirks.pdf|title=The Stone Representation Theorem for Boolean Algebras|last=Dirks|first=Matthew|date=|website=math.uchicago.edu|archive-url=|archive-date=|access-date=2019-12-08}}
• : A variant, Stone's representation theorem for distributive lattices, states that every distributive lattice is isomorphic to a sublattice of the power set lattice of some set.
• : Another variant, Stone's duality, 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.{{Cite journal|last=Schneider|first=Friedrich Martin|date=November 2017|title=A uniform Birkhoff theorem|journal=Algebra Universalis|volume=78|issue=3|pages=337–354|doi=10.1007/s00012-017-0460-1|issn=0002-5240|arxiv=1510.03166|s2cid=253600065 }}
• In the study of semigroups, the Wagner–Preston theorem provides a representation of an inverse semigroup S, as a homomorphic image of the set of partial bijections on S, and the semigroup operation given by composition.

• 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.{{nlab|id=Freyd-Mitchell+embedding+theorem|title=Freyd–Mitchell embedding theorem|access-date=2019-12-08}}
• 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.

• 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 states that a Hilbert space, such as the square-integrable function space L²(X) on a manifold X, any linear functional F is equal to the inner product with a fixed element $a\backslash in\; H$, i.e. $F(v)\; =\; \backslash langle\; a,v\backslash rangle$ for all $v\backslash in\; H$. The more general Riesz–Markov–Kakutani representation theorem has several versions, one of them identifying the dual space of C₀(X) with the set of regular measures on X.

• The Whitney embedding theorems embed any abstract manifold in some Euclidean space.
• The Nash embedding theorem embeds an abstract Riemannian manifold isometrically in a Euclidean space.{{Cite web|url=https://terrytao.wordpress.com/2016/05/11/notes-on-the-nash-embedding-theorem/|title=Notes on the Nash embedding theorem|date=2016-05-11|website=What's new|language=en|access-date=2019-12-08}}

• It is a basic result that every partially ordered set is isomorphic to a collection of sets ordered by inclusion (containment).

• A preference representation theorem states conditions for the existence of a utility function representing a preference relation. Examples are Von Neumann–Morgenstern utility theorem and Debreu's representation theorems.

• {{annotated link|Classification theorem}}

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Category:Mathematical theorems