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In mathematics, a representation theorem is a theorem that states that every abstract structure with certain properties is isomorphic to another (abstract or 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 ''C''0(''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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Representation theorem」の詳細全文を読む スポンサード リンク
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