CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> finite-dimensional vector space (Rev #10) in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="noindex,nofollow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> finite-dimensional vector space (Rev #10) </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/16628/#Item_2" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='linear_algebra'>Linear algebra</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/homotopy+theory'>homotopy theory</a>, <a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a>, <a class='existingWikiWord' href='/nlab/show/homotopy+type+theory'>homotopy type theory</a></strong></p> <p>flavors: <a class='existingWikiWord' href='/nlab/show/stable+homotopy+theory'>stable</a>, <a class='existingWikiWord' href='/nlab/show/equivariant+homotopy+theory'>equivariant</a>, <a class='existingWikiWord' href='/nlab/show/rational+homotopy+theory'>rational</a>, <a class='existingWikiWord' href='/nlab/show/p-adic+homotopy+theory'>p-adic</a>, <a class='existingWikiWord' href='/nlab/show/proper+homotopy+theory'>proper</a>, <a class='existingWikiWord' href='/nlab/show/geometric+homotopy+type+theory'>geometric</a>, <a class='existingWikiWord' href='/nlab/show/cohesive+homotopy+theory'>cohesive</a>, <a class='existingWikiWord' href='/nlab/show/directed+homotopy+theory'>directed</a>…</p> <p>models: <a class='existingWikiWord' href='/nlab/show/topological+homotopy+theory'>topological</a>, <a class='existingWikiWord' href='/nlab/show/simplicial+homotopy+theory'>simplicial</a>, <a class='existingWikiWord' href='/nlab/show/localic+homotopy+theory'>localic</a>, …</p> <p>see also <strong><a class='existingWikiWord' href='/nlab/show/algebraic+topology'>algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/Introduction+to+Topology+--+2'>Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Introduction+to+Homotopy+Theory'>Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/geometry+of+physics+--+homotopy+types'>geometry of physics -- homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/homotopy'>homotopy</a>, <a class='existingWikiWord' href='/nlab/show/higher+homotopy'>higher homotopy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/homotopy+type'>homotopy type</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Pi-algebra'>Pi-algebra</a>, <a class='existingWikiWord' href='/nlab/show/spherical+object'>spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/homotopy+coherent+category+theory'>homotopy coherent category theory</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/homotopical+category'>homotopical category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/model+category'>model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/category+of+fibrant+objects'>category of fibrant objects</a>, <a class='existingWikiWord' href='/nlab/show/cofibration+category'>cofibration category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Waldhausen+category'>Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/homotopy+category'>homotopy category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/Ho%28Top%29'>Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category'>(∞,1)-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/homotopy+category+of+an+%28infinity%2C1%29-category'>homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/homotopy'>left homotopy</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/cylinder+object'>cylinder object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/mapping+cone'>mapping cone</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/homotopy'>right homotopy</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/path+space+object'>path object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/mapping+cocone'>mapping cocone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/generalized+universal+bundle'>universal bundle</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/interval+object'>interval object</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/localization+at+geometric+homotopies'>homotopy localization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/infinitesimal+interval+object'>infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/homotopy+group'>homotopy group</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/fundamental+group'>fundamental group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/fundamental+group+of+a+topos'>fundamental group of a topos</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Brown-Grossman+homotopy+group'>Brown-Grossman homotopy group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/categorical+homotopy+groups+in+an+%28infinity%2C1%29-topos'>categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/geometric+homotopy+groups+in+an+%28infinity%2C1%29-topos'>geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-groupoid'>fundamental ∞-groupoid</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/fundamental+groupoid'>fundamental groupoid</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/path+groupoid'>path groupoid</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-groupoid+in+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-groupoid+of+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/fundamental+%28infinity%2C1%29-category'>fundamental (∞,1)-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/fundamental+category'>fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/fundamental+group+of+the+circle+is+the+integers'>fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/fundamental+theorem+of+covering+spaces'>fundamental theorem of covering spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Blakers-Massey+theorem'>Blakers-Massey theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/higher+homotopy+van+Kampen+theorem'>higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/nerve+theorem'>nerve theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Whitehead+theorem'>Whitehead's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Hurewicz+theorem'>Hurewicz theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Galois+theory'>Galois theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/homotopy+hypothesis'>homotopy hypothesis</a>-theorem</p> </li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#definition'>Definition</a></li><li><a href='#properties'>Properties</a><ul><li><a href='#CompactClosure'>Compact closure</a></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='definition'>Definition</h2> <p>A <a class='existingWikiWord' href='/nlab/show/vector+space'>vector space</a> is <em>finite-dimensional</em> if it admits a <a class='existingWikiWord' href='/nlab/show/finite+set'>finite</a> <a class='existingWikiWord' href='/nlab/show/basis'>basis</a>: if there exists a <a class='existingWikiWord' href='/nlab/show/natural+number'>natural number</a> <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> such that the vector space admits a <a class='existingWikiWord' href='/nlab/show/linear+isomorphism'>linear isomorphism</a> to the <a class='existingWikiWord' href='/nlab/show/direct+sum'>direct sum</a> of <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> copies of the <a class='existingWikiWord' href='/nlab/show/forgetful+functor'>underlying</a> vector space of the <a class='existingWikiWord' href='/nlab/show/ground+ring'>ground field</a>.</p> <p>The category <a class='existingWikiWord' href='/nlab/show/FinDimVect'>FinDimVect</a> of finite-dimensional vector spaces is of course the <a class='existingWikiWord' href='/nlab/show/full+subcategory'>full subcategory</a> of <a class='existingWikiWord' href='/nlab/show/Vect'>Vect</a> whose objects are finite-dimensional.</p> <h2 id='properties'>Properties</h2> <h3 id='CompactClosure'>Compact closure</h3> <div class='num_prop'> <h6 id='proposition'>Proposition</h6> <p>A vector space <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> is finite-dimensional precisely if the <a class='existingWikiWord' href='/nlab/show/hom-functor'>hom-functor</a> <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>hom</mi><mo stretchy='false'>(</mo><mi>V</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>:</mo><mi>Vect</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>\hom(V, -): Vect \to Set</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/preserved+limit'>preserves</a> <a class='existingWikiWord' href='/nlab/show/filtered+limit'>filtered colimits</a>.</p> </div> <div class='proof'> <h6 id='proof'>Proof</h6> <p>Every vector space <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>W</mi></mrow><annotation encoding='application/x-tex'>W</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/filtered+limit'>filtered colimit</a> of the <a class='existingWikiWord' href='/nlab/show/diagram'>diagram</a> of finite-dimensional subspaces <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>W</mi><mo>′</mo><mo>⊆</mo><mi>W</mi></mrow><annotation encoding='application/x-tex'>W' \subseteq W</annotation></semantics></math> and inclusions between them; applying this to <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>W</mi><mo>=</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>W = V</annotation></semantics></math>, the condition that <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>hom</mi><mo stretchy='false'>(</mo><mi>V</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\hom(V,-)</annotation></semantics></math> preserves filtered colimits implies that the canonical comparison map</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>colim</mi> <mrow><mi>fd</mi><mspace width='thickmathspace'></mspace><mi>V</mi><mo>′</mo><mo>⊆</mo><mi>V</mi></mrow></msub><mi>hom</mi><mo stretchy='false'>(</mo><mi>V</mi><mo>,</mo><mi>V</mi><mo>′</mo><mo stretchy='false'>)</mo><mo>→</mo><mi>hom</mi><mo stretchy='false'>(</mo><mi>V</mi><mo>,</mo><mi>V</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>colim_{fd\; V' \subseteq V} \hom(V, V') \to \hom(V, V)</annotation></semantics></math></div> <p>is an <a class='existingWikiWord' href='/nlab/show/isomorphism'>isomorphism</a>, so some element <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>f</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[f]</annotation></semantics></math> in the colimit represented by <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>V</mi><mo>→</mo><mi>V</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>f: V \to V'</annotation></semantics></math> gets mapped to <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mn>1</mn> <mi>V</mi></msub></mrow><annotation encoding='application/x-tex'>1_V</annotation></semantics></math>, i.e., <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>∘</mo><mi>f</mi><mo>=</mo><msub><mn>1</mn> <mi>V</mi></msub></mrow><annotation encoding='application/x-tex'>i \circ f = 1_V</annotation></semantics></math> for some inclusion <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>:</mo><mi>V</mi><mo>′</mo><mo>↪</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>i: V' \hookrightarrow V</annotation></semantics></math>. This implies <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math> is an isomorphism, so that <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> is finite-dimensional.</p> <p>In the converse direction, observe that <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>hom</mi><mo stretchy='false'>(</mo><mi>V</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\hom(V, -)</annotation></semantics></math> has a <a class='existingWikiWord' href='/nlab/show/right+adjoint'>right adjoint</a> (and in particular preserves filtered colimits) if <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> is finite-dimensional.</p> <p>To see this, first notice that the <a class='existingWikiWord' href='/nlab/show/dual+vector+space'>dual vector space</a> <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>V^\ast</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/functional'>functionals</a> <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>V</mi><mo>→</mo><mi>k</mi></mrow><annotation encoding='application/x-tex'>f: V \to k</annotation></semantics></math> to the <a class='existingWikiWord' href='/nlab/show/ground+ring'>ground field</a> is a <a class='existingWikiWord' href='/nlab/show/dualizable+object'>dual object</a> to <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> in the <a class='existingWikiWord' href='/nlab/show/monoidal+category'>monoidal category</a> sense, so that there is a <a class='existingWikiWord' href='/nlab/show/unit+of+an+adjunction'>counit</a> <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>eva</mi><mo lspace='verythinmathspace'>:</mo><msup><mi>V</mi> <mo>*</mo></msup><mo>⊗</mo><mi>V</mi><mo>→</mo><mi>k</mi></mrow><annotation encoding='application/x-tex'>eva \colon V^\ast \otimes V \to k</annotation></semantics></math> taking <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>⊗</mo><mi>v</mi><mo>↦</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>v</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f \otimes v \mapsto f(v)</annotation></semantics></math>. The <a class='existingWikiWord' href='/nlab/show/unit+of+an+adjunction'>unit</a> is uniquely determined from this counit and can be described using any basis <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>e</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>e_i</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> and dual basis <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mi>j</mi></msub></mrow><annotation encoding='application/x-tex'>f_j</annotation></semantics></math> as the map</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>→</mo><mi>V</mi><mo>⊗</mo><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>k \to V \otimes V^\ast</annotation></semantics></math></div> <p>taking <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo>↦</mo><msub><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mi>i</mi></msub><msub><mi>e</mi> <mi>i</mi></msub><mo>⊗</mo><msub><mi>f</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>1 \mapsto \sum_i e_i \otimes f_i</annotation></semantics></math>. We thus have an adjunction <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><msub><mo>⊗</mo> <mi>k</mi></msub><mi>V</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace'></mspace><mo>⊣</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>⊗</mo><msup><mi>V</mi> <mo>*</mo></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(- \otimes_k V) \; \dashv (- \otimes V^\ast)</annotation></semantics></math>, which is <a class='existingWikiWord' href='/nlab/show/mate'>mated</a> to an <a class='existingWikiWord' href='/nlab/show/adjunction'>adjunction</a> <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>hom</mi><mo stretchy='false'>(</mo><mi>V</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>⊣</mo><mi>hom</mi><mo stretchy='false'>(</mo><msup><mi>V</mi> <mo>*</mo></msup><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\hom(V, -) \dashv \hom(V^\ast, -)</annotation></semantics></math> by familiar hom-tensor adjunctions; thus <math class='maruku-mathml' display='inline' id='mathml_22c327f9e9cc4841133c82a2c9349370799a7d70_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>hom</mi><mo stretchy='false'>(</mo><mi>V</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\hom(V, -)</annotation></semantics></math> has a right adjoint.</p> </div> <p>This means that</p> <div class='num_prop'> <h6 id='proposition_2'>Proposition</h6> <p>Finite-dimensional vector spaces are exactly the <a class='existingWikiWord' href='/nlab/show/compact+object'>compact objects</a> of <a class='existingWikiWord' href='/nlab/show/Vect'>Vect</a> in the sense of <a class='existingWikiWord' href='/nlab/show/locally+presentable+category'>locally presentable categories</a>, but also the compact = <a class='existingWikiWord' href='/nlab/show/dualizable+object'>dualizable objects</a> in the sense of <a class='existingWikiWord' href='/nlab/show/monoidal+category'>monoidal category</a> theory. In particular the category <a class='existingWikiWord' href='/nlab/show/FinDimVect'>FinDimVect</a> is a <a class='existingWikiWord' href='/nlab/show/compact+closed+category'>compact closed category</a>.</p> </div> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/finite-dimensional+Hilbert+space'>finite-dimensional Hilbert space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/finite+abelian+category'>finite abelian category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Schur+functor'>Schur functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/quantum+information+theory+via+dagger-compact+categories'>finite quantum mechanics in terms of dagger-compact categories</a></p> </li> </ul> <h2 id='references'>References</h2> <p>Discussion of finite-dimensional vector spaces as <a class='existingWikiWord' href='/nlab/show/categorical+semantics'>categorical semantics</a> for <a class='existingWikiWord' href='/nlab/show/linear+logic'>linear logic</a>/<a class='existingWikiWord' href='/nlab/show/linear+type+theory'>linear type theory</a>:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/Beno%C3%AEt+Valiron'>Benoît Valiron</a>, <a class='existingWikiWord' href='/nlab/show/Steve+Zdancewic'>Steve Zdancewic</a>, <em>Finite Vector Spaces as Model of Simply-Typed Lambda-Calculi</em>, in: <em>Proc. of ICTAC’14</em>, Lecture Notes in Computer Science <strong>8687</strong>, Springer (2014) 442-459 [[arXiv:1406.1310](https://arxiv.org/abs/1406.1310), <a href='https://doi.org/10.1007/978-3-319-10882-7_26'>doi:10.1007/978-3-319-10882-7_26</a>, slides:<a href='https://www.cs.bham.ac.uk/~drg/bll/steve.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/Zdancewic-LinearLogicAlgebra.pdf' title='pdf'>pdf</a>]</p> <blockquote> <p>(focus on <a class='existingWikiWord' href='/nlab/show/finite+field'>finite</a> <a class='existingWikiWord' href='/nlab/show/ground+ring'>ground fields</a>)</p> </blockquote> </li> <li id='Murfet14'> <p><a class='existingWikiWord' href='/nlab/show/Daniel+Murfet'>Daniel Murfet</a>, <em>Logic and linear algebra: an introduction</em> [[arXiv:1407.2650](http://arxiv.org/abs/1407.2650)]</p> </li> </ul> <p> </p> </div>  <div class="revisedby"> <p> Revision on September 4, 2023 at 11:08:35 by <a href="/nlab/author/Urs+Schreiber" style="color: #005c19">Urs Schreiber</a> See the <a href="/nlab/history/finite-dimensional+vector+space" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="https://nforum.ncatlab.org/discussion/16628/#Item_2">Discuss</a><span class="backintime"><a href="/nlab/show/finite-dimensional+vector+space" accesskey="F" class="navlinkbackintime" id="to_next_revision" rel="nofollow">Next revision</a> (to current)</span><span class="backintime"><a href="/nlab/revision/finite-dimensional+vector+space/9" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a> (9 more)</span><a href="/nlab/show/finite-dimensional+vector+space" class="navlink" id="to_current_revision">Current version of page</a><a href="/nlab/revision/diff/finite-dimensional+vector+space/10" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/finite-dimensional+vector+space" accesskey="S" class="navlink" id="history" rel="nofollow">History (10 revisions)</a><a href="/nlab/rollback/finite-dimensional+vector+space?rev=10" class="navlink" id="rollback" rel="nofollow">Rollback</a> <a href="/nlab/revision/finite-dimensional+vector+space/10/cite" style="color: black">Cite</a> <a href="/nlab/source/finite-dimensional+vector+space/10" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>