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two Banach spaces isomorphic?</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/functional+analysis+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="measure_and_probability_theory">Measure and probability theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/measure+theory">measure theory</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/probability+theory">probability theory</a></strong></p> <p>(<a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a>)</p> <h2 id="measure_theory">Measure theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/measurable+space">measurable space</a>, <a class="existingWikiWord" href="/nlab/show/measurable+locale">measurable locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measure">measure</a>, <a class="existingWikiWord" href="/nlab/show/measure+space">measure space</a></p> </li> <li> <p><a 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<a class="existingWikiWord" href="/nlab/show/relative+entropy">relative entropy</a></p> </li> </ul> <h2 id="information_geometry">Information geometry</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/information+geometry">information geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/information+metric">information metric</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wasserstein+metric">Wasserstein metric</a></p> </li> </ul> <h2 id="thermodynamics">Thermodynamics</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second+law+of+thermodynamics">second law of thermodynamics</a>, <a class="existingWikiWord" href="/nlab/show/generalized+second+law+of+theormodynamics">generalized second law</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ergodic+theory">ergodic theory</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Riesz+representation+theorem">Riesz representation theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Finetti%27s+theorem">de Finetti's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/law+of+large+numbers">law of large numbers</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kolmogorov+extension+theorem">Kolmogorov extension theorem</a></p> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/machine+learning">machine learning</a>, <a class="existingWikiWord" href="/nlab/show/neural+networks">neural networks</a></li> </ul> </div></div> </div> </div> <h1 id="the_riesz_representation_theorems">The Riesz representation theorems</h1> <div class='maruku_toc'> <ul> <li><a href='#summary'>Summary</a></li> <li><a href='#'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>C</mi> <mi>c</mi> <mrow><mo>*</mo><mo lspace="verythinmathspace" rspace="0em">+</mo></mrow></msubsup><mo>=</mo><msup><mover><mi>RM</mi><mo>¯</mo></mover> <mo>+</mo></msup></mrow><annotation encoding="application/x-tex">C_c^{{*}{+}} = \overline{RM}^+</annotation></semantics></math></a></li> <ul> <li><a href='#theorem_riesz'>Theorem (Riesz)</a></li> </ul> <li><a href='#_2'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>C</mi> <mn>0</mn> <mo>*</mo></msubsup><mo>=</mo><mi>RM</mi></mrow><annotation encoding="application/x-tex">C_0^* = RM</annotation></semantics></math></a></li> <ul> <li><a href='#theorem_rieszmarkov'>Theorem (Riesz–Markov)</a></li> </ul> <li><a href='#_3'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>L</mi> <mn>1</mn> <mo>*</mo></msubsup><mo>=</mo><msub><mi>L</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">L_1^* = L_0</annotation></semantics></math></a></li> <li><a href='#_4'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>L</mi> <mi>p</mi> <mo>*</mo></msubsup><mo>=</mo><msub><mi>L</mi> <mrow><mn>1</mn><mo>−</mo><mi>p</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_p^* = L_{1 - p}</annotation></semantics></math></a></li> <li><a href='#_5'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>*</mo></msup><mo>=</mo><mover><mi>H</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">H^* = \bar{H}</annotation></semantics></math></a></li> <li><a href='#_6'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>L</mi> <mn>0</mn> <mo>*</mo></msubsup><mo>=</mo><mi>BA</mi></mrow><annotation encoding="application/x-tex">L_0^* = BA</annotation></semantics></math></a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="summary">Summary</h2> <p>There are various related theorems in <a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a> and <a class="existingWikiWord" href="/nlab/show/measure+theory">measure theory</a> stating, under appropriate conditions, that the <a class="existingWikiWord" href="/nlab/show/dual+vector+space">topological linear duals</a> of various familiar <a class="existingWikiWord" href="/nlab/show/Banach+spaces">Banach spaces</a> (or something similar) are other familiar Banach spaces. Most of these are due in part to <a class="existingWikiWord" href="/nlab/show/Frigyes+Riesz">Frigyes Riesz</a>, and many of them are named after him. Here we will consider them all together.</p> <p>Throughout, we use notation for <a class="existingWikiWord" href="/nlab/show/integrals">integrals</a> in which unnecessary ‘<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">d</mi></mrow><annotation encoding="application/x-tex">\mathrm{d}</annotation></semantics></math>’s are dropped; see the discussion on notation at <em><a class="existingWikiWord" href="/nlab/show/measure+space">measure space</a></em>.</p> <h2 id=""><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>C</mi> <mi>c</mi> <mrow><mo>*</mo><mo lspace="verythinmathspace" rspace="0em">+</mo></mrow></msubsup><mo>=</mo><msup><mover><mi>RM</mi><mo>¯</mo></mover> <mo>+</mo></msup></mrow><annotation encoding="application/x-tex">C_c^{{*}{+}} = \overline{RM}^+</annotation></semantics></math></h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/locally+compact+Hausdorff+space">locally compact Hausdorff space</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>c</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_c(X)</annotation></semantics></math> be the space of <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (valued in the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>) with <a class="existingWikiWord" href="/nlab/show/compact+subset">compact</a> <a class="existingWikiWord" href="/nlab/show/support">support</a>; make <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>c</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_c(X)</annotation></semantics></math> into a <a class="existingWikiWord" href="/nlab/show/locally+convex+space">locally convex space</a> with the topology of <span class="newWikiWord">uniform convergence on compact subsets<a href="/nlab/new/uniform+convergence+topology">?</a></span>; the <a class="existingWikiWord" href="/nlab/show/dual+vector+space">dual vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>c</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">C_c(X)*</annotation></semantics></math> of this is (of course) the space of <a class="existingWikiWord" href="/nlab/show/continuous+map">continuous</a> <a class="existingWikiWord" href="/nlab/show/linear+functional">linear functional</a>s on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>c</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_c(X)</annotation></semantics></math>; and the <a class="existingWikiWord" href="/nlab/show/positive+cone">positive cone</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>c</mi></msub><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mrow><mo>*</mo><mo lspace="verythinmathspace" rspace="0em">+</mo></mrow></msup></mrow><annotation encoding="application/x-tex">C_c(X)^{{*}{+}}</annotation></semantics></math> of this is the space of <a class="existingWikiWord" href="/nlab/show/positive+linear+functional">positive linear functional</a>s on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>c</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_c(X)</annotation></semantics></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RM</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">RM(X)</annotation></semantics></math> be the space of <a class="existingWikiWord" href="/nlab/show/finite+measure">finite</a> <a class="existingWikiWord" href="/nlab/show/Radon+measure">Radon measure</a>s on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>; make <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RM</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">RM(X)</annotation></semantics></math> into a <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a> with the <span class="newWikiWord">total variation<a href="/nlab/new/total+variation">?</a></span> norm; the <a class="existingWikiWord" href="/nlab/show/extended+positive+cone">extended positive cone</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mover><mrow><mi>RM</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mo>¯</mo></mover> <mo>+</mo></msup></mrow><annotation encoding="application/x-tex">\overline{RM(X)}^+</annotation></semantics></math> of this is the space of <a class="existingWikiWord" href="/nlab/show/positive+measure">positive</a> Radon measures on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. <a class="existingWikiWord" href="/nlab/show/integral">Integration</a> gives a map from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mover><mrow><mi>RM</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mo>¯</mo></mover> <mo>+</mo></msup></mrow><annotation encoding="application/x-tex">\overline{RM(X)}^+</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>c</mi></msub><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mrow><mo>*</mo><mo lspace="verythinmathspace" rspace="0em">+</mo></mrow></msup></mrow><annotation encoding="application/x-tex">C_c(X)^{{*}{+}}</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>↦</mo><mo stretchy="false">(</mo><mi>f</mi><mo>↦</mo><msub><mo>∫</mo> <mi>X</mi></msub><mi>f</mi><mi>μ</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mu \mapsto (f \mapsto \int_X f \mu) .</annotation></semantics></math></div> <div class="num_theorem" id="Cc"> <h6 id="theorem_riesz">Theorem (Riesz)</h6> <p>This map is a <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>c</mi></msub><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mrow><mo>*</mo><mo lspace="verythinmathspace" rspace="0em">+</mo></mrow></msup><mo>≅</mo><msup><mover><mrow><mi>RM</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mo>¯</mo></mover> <mo>+</mo></msup><mo>.</mo></mrow><annotation encoding="application/x-tex"> C_c(X)^{{*}{+}} \cong \overline{RM(X)}^+ .</annotation></semantics></math></div></div> <h2 id="_2"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>C</mi> <mn>0</mn> <mo>*</mo></msubsup><mo>=</mo><mi>RM</mi></mrow><annotation encoding="application/x-tex">C_0^* = RM</annotation></semantics></math></h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/locally+compact+Hausdorff+space">locally compact Hausdorff space</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_0(X)</annotation></semantics></math> be the space of <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (valued in the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>) on the <a class="existingWikiWord" href="/nlab/show/one-point+compactification">one-point compactification</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (so vanishing ‘at infinity’); make <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_0(X)</annotation></semantics></math> into a <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a> with the <a class="existingWikiWord" href="/nlab/show/supremum+norm">supremum norm</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RM</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">RM(X)</annotation></semantics></math> be the space of <a class="existingWikiWord" href="/nlab/show/finite+measure">finite</a> <a class="existingWikiWord" href="/nlab/show/Radon+measure">Radon measure</a>s on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>; make <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RM</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">RM(X)</annotation></semantics></math> into a <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a> with the <span class="newWikiWord">total variation<a href="/nlab/new/total+variation">?</a></span> norm. <a class="existingWikiWord" href="/nlab/show/integral">Integration</a> gives a map from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RM</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">RM(X)</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/dual+vector+space">dual vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C_0(X)^*</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_0(X)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>↦</mo><mo stretchy="false">(</mo><mi>f</mi><mo>↦</mo><msub><mo>∫</mo> <mi>X</mi></msub><mi>f</mi><mi>μ</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mu \mapsto (f \mapsto \int_X f \mu) .</annotation></semantics></math></div> <div class="num_theorem" id="C0"> <h6 id="theorem_rieszmarkov">Theorem (Riesz–Markov)</h6> <p>This map is an <a class="existingWikiWord" href="/nlab/show/isometric+isomorphism">isometric isomorphism</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>≅</mo><mi>RM</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> C_0(X)^* \cong RM(X) .</annotation></semantics></math></div></div> <h2 id="_3"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>L</mi> <mn>1</mn> <mo>*</mo></msubsup><mo>=</mo><msub><mi>L</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">L_1^* = L_0</annotation></semantics></math></h2> <h2 id="_4"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>L</mi> <mi>p</mi> <mo>*</mo></msubsup><mo>=</mo><msub><mi>L</mi> <mrow><mn>1</mn><mo>−</mo><mi>p</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_p^* = L_{1 - p}</annotation></semantics></math></h2> <h2 id="_5"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>*</mo></msup><mo>=</mo><mover><mi>H</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">H^* = \bar{H}</annotation></semantics></math></h2> <h2 id="_6"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>L</mi> <mn>0</mn> <mo>*</mo></msubsup><mo>=</mo><mi>BA</mi></mrow><annotation encoding="application/x-tex">L_0^* = BA</annotation></semantics></math></h2> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></li> </ul> <h2 id="references">References</h2> <p>A proof of Theorem <a class="maruku-ref" href="#Cc"></a> in <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a> (in the case where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/compactum">compactum</a>) is given in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Thierry+Coquand">Thierry Coquand</a>, <a class="existingWikiWord" href="/nlab/show/Bas+Spitters">Bas Spitters</a>, <em>Integrals and Valuations</em> (<a href="http://arxiv.org/abs/0808.1522">arXiv:0808.1522</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 11, 2017 at 16:24:15. 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