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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/11448/#Item_1" 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"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <blockquote> <p>For the <a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a> of the same name see at <a class="existingWikiWord" href="/nlab/show/Smith+space+%28generalized+smooth+space%29">Smith space (generalized smooth space)</a>.</p> </blockquote> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <h2 id="definition">Definition</h2> <p>In the context of <a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a>, a <strong>Smith space</strong> is a <a class="existingWikiWord" href="/nlab/show/complete+topological+vector+space">complete</a> <a class="existingWikiWord" href="/nlab/show/locally+convex+topological+vector+space">locally convex</a> topological <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>-vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> that admits a <a class="existingWikiWord" href="/nlab/show/compact+object">compact</a> <a class="existingWikiWord" href="/nlab/show/absolutely+convex">absolutely convex</a> subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>⊂</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">K \subset V</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">⋃</mo> <mrow><mi>c</mi><mo>></mo><mn>0</mn></mrow></msub><mi>c</mi><mi>K</mi></mrow><annotation encoding="application/x-tex">V = \bigcup_{c \gt 0} c K</annotation></semantics></math> with the induced <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+space">compactly generated topology</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>.</p> <h2 id="examples">Examples</h2> <ul> <li>For a compact absolutely convex set, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, the space of <a class="existingWikiWord" href="/nlab/show/measures">measures</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M(S)</annotation></semantics></math>, with the compactly generated topology is a Smith space.</li> </ul> <h2 id="properties">Properties</h2> <p>The category of Smith spaces is <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalent</a> to the opposite of the category of <a class="existingWikiWord" href="/nlab/show/Banach+spaces">Banach spaces</a>. More precisely, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is a Banach space, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(V,\mathbb{R})</annotation></semantics></math> is a Smith space; and if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> is a Smith space, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(W,\mathbb{R})</annotation></semantics></math> is a Banach space, where in both cases we endow the dual space with the compact-open topology. The corresponding biduality maps are isomorphisms.</p> <p>Smith spaces are more natural than Banach spaces from the perspective of <a class="existingWikiWord" href="/nlab/show/condensed+mathematics">condensed mathematics</a> because they are controlled by a nice compact subset instead of a nice open subset (the unit ball) as in the Banach setting. Also, Banach spaces are <a class="existingWikiWord" href="/nlab/show/filtered+colimits">filtered colimits</a> of Smith spaces while Smith spaces are <a class="existingWikiWord" href="/nlab/show/filtered+limits">filtered limits</a> of Banach spaces. Filtered colimits have better algebraic and homological properties.</p> <p>Smith spaces embed fully faithfully in condensed R-vector spaces, but Smith spaces do not form an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>. There is an induced functor from Waelbrock’s enlarged (abelian) category of quotients of Smith spaces, but that functor is not fully faithful.</p> <p>Smith spaces are the same as Waelbrock dual spaces.</p> <p>A Banach space is a Smith space if and only if it is finite-dimensional.</p> <h2 id="references">References</h2> <p>The original treatment is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Marianne+Smith">Marianne Smith</a>, <em>The Pontrjagin duality theorem in linear spaces</em>, Annals of Mathematics. 56(2): 248–253, (<a href="https://doi.org/10.2307%2F1969798">doi:10.2307/1969798</a>).</li> </ul> <p>In the context of <a class="existingWikiWord" href="/nlab/show/condensed+mathematics">condensed mathematics</a>, see</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Peter+Scholze">Peter Scholze</a>, <em>Lectures on Analytic Geometry</em>, (<a href="https://www.math.uni-bonn.de/people/scholze/Analytic.pdf">pdf</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 16, 2020 at 18:43:38. 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