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- Representation_theorem abstract "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. Another variant, 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.in category theory, 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.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.in functional analysis 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 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 in some Euclidean space. The Nash embedding theorem embeds an abstract Riemannian manifold isometrically in an Euclidean space.".
- Representation_theorem wikiPageID "4783135".
- Representation_theorem wikiPageRevisionID "600967290".
- Representation_theorem hasPhotoCollection Representation_theorem.
- Representation_theorem subject Category:Mathematical_theorems.
- Representation_theorem type Abstraction100002137.
- Representation_theorem type Communication100033020.
- Representation_theorem type MathematicalTheorems.
- Representation_theorem type Message106598915.
- Representation_theorem type Proposition106750804.
- Representation_theorem type Statement106722453.
- Representation_theorem type Theorem106752293.
- Representation_theorem comment "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.".
- Representation_theorem label "Representation theorem".
- Representation_theorem label "Teorema de representación".
- Representation_theorem sameAs Teorema_de_representación.
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- Representation_theorem sameAs Q7314222.
- Representation_theorem sameAs Q7314222.
- Representation_theorem sameAs Representation_theorem.
- Representation_theorem wasDerivedFrom Representation_theorem?oldid=600967290.
- Representation_theorem isPrimaryTopicOf Representation_theorem.