Representation theorem

<h2 id="examples">Examples</h2>
<h3>Algebra</h3>
<ul><li><a href="/facts/Cayley%27s_theorem/a0M460K8">Cayley's theorem</a> states that every <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> is <a href="/facts/Group_isomorphism/LC4N1aDw">isomorphic</a> to a <a href="/facts/Permutation_group/vRsNNwwx">permutation group</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a></li>
<li><a href="/facts/Representation_theory/3ydLtaGH">Representation theory</a> studies properties of abstract groups via their <a href="/facts/Group_representation/FsuuDagQ">representations</a> as <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformations</a> of <a href="/facts/Vector_space/M1pxMLx2">vector spaces</a>.</li>
<li><a href="/facts/Stone%27s_representation_theorem/tN870P19">Stone's representation theorem</a> for <a href="/facts/Boolean_algebra_(structure)/cHbo65jm">Boolean algebras</a> states that every Boolean algebra is isomorphic to a <a href="/facts/Field_of_sets/4E3vQl4f">field of sets</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
A variant, <a href="/facts/Stone%27s_representation_theorem_for_distributive_lattices/sHczUFeb">Stone's representation theorem for distributive lattices</a>, states that every <a href="/facts/Distributive_lattice/97DFz1cu">distributive lattice</a> is isomorphic to a sublattice of the <a href="/facts/Power_set/FVuf8PDD">power set</a> lattice of some set.
Another variant, <a href="/facts/Stone%27s_duality/w1QqNX39">Stone's duality</a>, states that there exists a duality (in the sense of an arrow-reversing equivalence) between the <a href="/facts/Category_(mathematics)/7WzxUThf">categories</a> of Boolean algebras and that of <a href="/facts/Stone_space/wqW5xc48">Stone spaces</a>.</li>
<li>The <a href="/facts/Poincar%C3%A9%E2%80%93Birkhoff%E2%80%93Witt_theorem/YSGbvgQv">Poincaré–Birkhoff–Witt theorem</a> states that every <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> embeds into the commutator Lie algebra of its <a href="/facts/Universal_enveloping_algebra/VJ3Syz4W">universal enveloping algebra</a>.</li>
<li><a href="/facts/Ado%27s_theorem/sPoJfDNR">Ado's theorem</a> states that every finite-dimensional <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> of <a href="/facts/Characteristic_zero/D2VzcQaG">characteristic zero</a> embeds into the Lie algebra of <a href="/facts/Endomorphism/s7NNlSea">endomorphisms</a> of some finite-dimensional vector space.</li>
<li><a href="/facts/Birkhoff%27s_HSP_theorem/vXWYFyDI">Birkhoff's HSP theorem</a> states that every <a href="/facts/Model_(model_theory)/GsMx1fEb">model</a> of an algebra <i>A</i> is the homomorphic image of a <a href="/facts/Subalgebra/9PEHNF5r">subalgebra</a> of a <a href="/facts/Direct_product/GehGQmNV">direct product</a> of copies of <i>A</i>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></li>
<li>In the study of <a href="/facts/Semigroup/rgSdk2kt">semigroups</a>, the <a href="/facts/Wagner%E2%80%93Preston_theorem/gz4ezaRq">Wagner–Preston theorem</a> provides a representation of an <a href="/facts/Inverse_semigroup/gz4ezaRq">inverse semigroup</a> <i>S</i>, as a homomorphic image of the set of <a href="/facts/Partial_bijection/j4gTuTmW">partial bijections</a> on <i>S</i>, and the semigroup operation given by <a href="/facts/Composition_of_relations/tD8joBlo">composition</a>.</li></ul>
<h3>Category theory</h3>
<ul><li>The <a href="/facts/Yoneda_lemma/j04vXd6x">Yoneda lemma</a> provides a <a href="/facts/Full_and_faithful_functors/bMM75v7e">full and faithful</a> <a href="/facts/Limit_(category_theory)/JoLugLs3">limit</a>-preserving embedding of any category into a category of <a href="/facts/Presheaf/MM3hcRw4">presheaves</a>.</li>
<li><a href="/facts/Mitchell%27s_embedding_theorem/6Mmx0k9e">Mitchell's embedding theorem</a> for <a href="/facts/Abelian_categories/4WN473Yc">abelian categories</a> realises every <a href="/facts/Small_category/7WzxUThf">small</a> abelian category as a full (and exactly embedded) <a href="/facts/Subcategory/k4lEeDDb">subcategory</a> of a <a href="/facts/Category_of_modules/AnnF9JKf">category of modules</a> over some <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></li>
<li><a href="/facts/Mostowski_collapse_lemma/nQISwDvn">Mostowski's collapsing theorem</a> states that every well-founded extensional structure is isomorphic to a transitive set with the ∈-relation.</li>
<li>One of the fundamental theorems in <a href="/facts/Sheaf_(mathematics)/MM3hcRw4">sheaf</a> theory states that every sheaf over a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> can be thought of as a sheaf of <a href="/facts/Section_(fiber_bundle)/DLnpXS5a">sections</a> of some (étalé) bundle over that space: the categories of sheaves on a topological space and that of <a href="/facts/%C3%89tal%C3%A9_space/MM3hcRw4">étalé spaces</a> over it are equivalent, where the equivalence is given by the <a href="/facts/Functor/l2u3rIBE">functor</a> that sends a bundle to its sheaf of (local) sections.</li></ul>
<h3>Functional analysis</h3>
<ul><li>The <a href="/facts/Gelfand%E2%80%93Naimark%E2%80%93Segal_construction/jm6BugY5">Gelfand–Naimark–Segal construction</a> embeds any <a href="/facts/C*-algebra/B3tnGHIo">C*-algebra</a> in an algebra of <a href="/facts/Bounded_operator/LYmDTIqR">bounded operators</a> on some <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a>.</li>
<li>The <a href="/facts/Gelfand_representation/QV54f7Jm">Gelfand representation</a> (also known as the commutative Gelfand–Naimark theorem) states that any commutative <a href="/facts/C*-algebra/B3tnGHIo">C*-algebra</a> is isomorphic to an algebra of continuous functions on its <a href="/facts/Gelfand_spectrum/QV54f7Jm">Gelfand spectrum</a>. It can also be seen as the construction as a duality between the category of commutative <a href="/facts/C*-algebras/B3tnGHIo">C*-algebras</a> and that of <a href="/facts/Compact_Hausdorff_space/d0cgXJH7">compact Hausdorff spaces</a>.</li>
<li>The <a href="/facts/Riesz_representation_theorem/9CjKDUII">Riesz representation theorem</a> states that a <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a>, such as the <a href="/facts/Square-integrable_function/IASblcJJ">square-integrable function</a> space <i>L</i>2(<i>X</i>) on a manifold <i>X</i>, any linear functional <i>F</i> is equal to the inner product with a fixed element 
  
    
      
        a
        ∈
        H
      
    
    {\displaystyle a\in H}
  
, i.e. 
  
    
      
        F
        (
        v
        )
        =
        ⟨
        a
        ,
        v
        ⟩
      
    
    {\displaystyle F(v)=\langle a,v\rangle }
  
 for all 
  
    
      
        v
        ∈
        H
      
    
    {\displaystyle v\in H}
  
. The more general <a href="/facts/Riesz%E2%80%93Markov%E2%80%93Kakutani_representation_theorem/vtJ7grhH">Riesz–Markov–Kakutani representation theorem</a> has several versions, one of them identifying the dual space of <i>C</i>0(<i>X</i>) with the set of regular measures on <i>X</i>.</li></ul>
<h3>Geometry</h3>
<ul><li>The <a href="/facts/Whitney_embedding_theorem/CyFVXalS">Whitney embedding theorems</a> embed any abstract <a href="/facts/Manifold/rXPERJtu">manifold</a> in some <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>.</li>
<li>The <a href="/facts/Nash_embedding_theorem/a7eLrRLM">Nash embedding theorem</a> embeds an abstract <a href="/facts/Riemannian_manifold/5KYnmEgF">Riemannian manifold</a> isometrically in a <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li></ul>
<h3>Order Theory</h3>
<ul><li>It is a basic result that every <a href="/facts/Partially_ordered_set/qb4a3FAX">partially ordered set</a> is <a href="/facts/Order_isomorphism/PwpVTjW5">isomorphic</a> to a collection of <a href="/facts/Set_theory/1ptfsvUP">sets</a> ordered by <a href="/facts/Subset/SJGERmbA">inclusion</a> (containment).</li></ul>
<h3>Economics</h3>
<ul><li>A <a href="/facts/Preference_representation_theorem/CrLAJQHz">preference representation theorem</a> states conditions for the existence of a <a href="/facts/Utility_function/wlMDG74L">utility function</a> representing a <a href="/facts/Preference_(economics)/4h4tND8S">preference relation</a>. Examples are <a href="/facts/Von_Neumann%E2%80%93Morgenstern_utility_theorem/jKfsUBQ0">Von Neumann–Morgenstern utility theorem</a> and <a href="/facts/Debreu%27s_representation_theorems/at0YJI45">Debreu's representation theorems</a>.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Classification_theorem/NWGgtNm6">Classification theorem</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>"Cayley's Theorem and its Proof". www.sjsu.edu. Retrieved 2019-12-08. <a href="http://www.sjsu.edu/faculty/watkins/cayleyth.htm" target="_blank">http://www.sjsu.edu/faculty/watkins/cayleyth.htm</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Dirks, Matthew. "The Stone Representation Theorem for Boolean Algebras" (PDF). math.uchicago.edu. Retrieved 2019-12-08. <a href="http://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Dirks.pdf" target="_blank">http://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Dirks.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Schneider, Friedrich Martin (November 2017). "A uniform Birkhoff theorem". Algebra Universalis. 78 (3): 337–354. arXiv:1510.03166. doi:10.1007/s00012-017-0460-1. ISSN 0002-5240. S2CID 253600065. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Freyd–Mitchell embedding theorem at the nLab <a href="https://ncatlab.org/nlab/show/Freyd-Mitchell+embedding+theorem" target="_blank">https://ncatlab.org/nlab/show/Freyd-Mitchell+embedding+theorem</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>"Notes on the Nash embedding theorem". What's new. 2016-05-11. Retrieved 2019-12-08. <a href="https://terrytao.wordpress.com/2016/05/11/notes-on-the-nash-embedding-theorem/" target="_blank">https://terrytao.wordpress.com/2016/05/11/notes-on-the-nash-embedding-theorem/</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
</ol>

Representation theorem open-in-new

Representation theorem