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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" 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> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="functional_analysis">Functional analysis</h4> <div class="hide"><div> <ul> <li><strong><a class="existingWikiWord" href="/nlab/show/functional+analysis">Functional Analysis</a></strong></li> </ul> <h2 id="overview_diagrams">Overview diagrams</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/TVS+relationships">topological vector spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/diagram+of+LCTVS+properties">locally convex topological vector spaces</a></p> </li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+convex+topological+vector+space">locally convex topological vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Banach+space">Banach Spaces</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/reflexive+Banach+space">reflexive</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Smith+space+%28functional+analysis%29">Smith Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert Spaces</a>, <a class="existingWikiWord" href="/nlab/show/Fr%C3%A9chet+space">Fréchet Spaces</a>, <a class="existingWikiWord" href="/nlab/show/Sobolev+space">Sobolev spaces</a>, <a class="existingWikiWord" href="/nlab/show/Lebesgue+space">Lebesgue Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bornological+vector+space">Bornological Vector Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/barrelled+topological+vector+space">Barrelled Vector Spaces</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+operator">linear operator</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/bounded+linear+operator">bounded</a>, <a class="existingWikiWord" href="/nlab/show/unbounded+linear+operator">unbounded</a>, <a class="existingWikiWord" href="/nlab/show/self-adjoint+operator">self-adjoint</a>, <a class="existingWikiWord" href="/nlab/show/compact+operator">compact</a>, <a class="existingWikiWord" href="/nlab/show/Fredholm+operator">Fredholm</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum+of+an+operator">spectrum of an operator</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebras">operator algebras</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/functional+calculus">functional calculus</a></li> </ul> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Stone-Weierstrass+theorem">Stone-Weierstrass theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+theory">spectral theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+theorem">spectral theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functional+calculus">functional calculus</a></p> </li> <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/measure+theory">measure theory</a></p> </li> </ul> <h2 id="topics_in_functional_analysis">Topics in Functional Analysis</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/basis+in+functional+analysis">Bases</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theories+in+functional+analysis">Algebraic Theories in Functional Analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/an+elementary+treatment+of+Hilbert+spaces">An Elementary Treatment of Hilbert Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isomorphism+classes+of+Banach+spaces">When are 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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#duality'>Duality</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#examples_and_counterexamples'>Examples and counterexamples</a></li> <li><a href='#references'>References</a></li> </ul> </div> <p>Throughout, we work in the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bant</mi></mrow><annotation encoding="application/x-tex">Bant</annotation></semantics></math> whose objects are Banach spaces and where the morphisms are continuous linear maps. References below to the unit ball suggest that it <em>might</em> be premature to cast everything in terms of a certain subcategory of TVS.</p> <h2 id="duality">Duality</h2> <p>Following the lead of Mac Lane (2nd ed.) Section IV.2, which does this for vector spaces) but with slight changes in notation: let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>D</mi><mo>¯</mo></mover><mo>:</mo><mi>Bant</mi><mo>→</mo><mi>Bant</mi></mrow><annotation encoding="application/x-tex">\overline{D}:Bant\to Bant</annotation></semantics></math> be the contravariant functor which takes a Banach space to its <a class="existingWikiWord" href="/nlab/show/dual+vector+space">dual space</a>, and sends a continuous linear map to its adjoint/dual map.</p> <p>This gives rise, in a straightforward way, to two functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>L</mi></msub><mo>:</mo><mi>Bant</mi><mo>→</mo><msup><mi>Bant</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">D_L: Bant \to Bant^{op}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>R</mi></msub><mo>:</mo><msup><mi>Bant</mi> <mi>op</mi></msup><mo>→</mo><mi>Bant</mi></mrow><annotation encoding="application/x-tex">D_R: Bant^{op}\to Bant</annotation></semantics></math>. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>L</mi></msub></mrow><annotation encoding="application/x-tex">D_L</annotation></semantics></math> is the left adjoint of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">D_R</annotation></semantics></math>, that is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Bant</mi> <mi>op</mi></msup><mo stretchy="false">(</mo><msub><mi>D</mi> <mi>L</mi></msub><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>Bant</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>D</mi> <mi>R</mi></msub><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Bant^{op}(D_L X, Y) \cong Bant(X, D_R Y) </annotation></semantics></math></div> <p>In general</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Bant</mi> <mi>op</mi></msup><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mi>D</mi> <mi>L</mi></msub><mi>X</mi><mo stretchy="false">)</mo><mo>¬</mo><mo>≅</mo><mi>Bant</mi><mo stretchy="false">(</mo><msub><mi>D</mi> <mi>R</mi></msub><mi>Y</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Bant^{op}(Y, D_L X) \not\cong Bant(D_R Y, X) </annotation></semantics></math></div> <p>so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>L</mi></msub></mrow><annotation encoding="application/x-tex">D_L</annotation></semantics></math> is not a right adjoint of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">D_R</annotation></semantics></math>. For example: take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> to be the ground field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> and <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> to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">c_0</annotation></semantics></math> with the usual supremum norm.</p> <p><strong>Not-a-proof-yet</strong> of this claim: we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">D_R(K)\cong K</annotation></semantics></math> and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bant</mi></mrow><annotation encoding="application/x-tex">Bant</annotation></semantics></math>-morphisms from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">c_0</annotation></semantics></math> are just vectors in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">c_0</annotation></semantics></math>; but the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bant</mi></mrow><annotation encoding="application/x-tex">Bant</annotation></semantics></math>-morphisms from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>L</mi></msub><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>≅</mo><msup><mi>ℓ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">D_L(c_0)\cong\ell^1</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> correspond to the vectors in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℓ</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">\ell^\infty</annotation></semantics></math>. (It would seem from this example that even in <a class="existingWikiWord" href="/nlab/show/dream+mathematics">dream mathematics</a> one doesn’t get <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>L</mi></msub></mrow><annotation encoding="application/x-tex">D_L</annotation></semantics></math> being a right adjoin of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">D_R</annotation></semantics></math>.)</p> <p><strong>Unit and counit.</strong> The unit of this adjunction is the canonical map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>κ</mi> <mi>X</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><mo stretchy="false">(</mo><msup><mi>X</mi> <mo>*</mo></msup><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\kappa_X: X\to (X^*)^*</annotation></semantics></math> from a Banach space <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> to its second dual <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mrow><mo>*</mo><mo>*</mo></mrow></msup></mrow><annotation encoding="application/x-tex">X^{**}</annotation></semantics></math>. In the presence of <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">Choice</a>, the <a class="existingWikiWord" href="/nlab/show/Hahn%E2%80%93Banach+theorem">Hahn–Banach theorem</a> ensures that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>κ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\kappa_X</annotation></semantics></math> is an isometry.</p> <p>To get things to run smoothly, we seem to need more than <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>κ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\kappa_X</annotation></semantics></math> being monic in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bant</mi></mrow><annotation encoding="application/x-tex">Bant</annotation></semantics></math>; but I (YC) am not sure which of the usual variants – extreme, regular, strong, strict – is the key one.</p> <p>The counit map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">veps</mo> <mi>X</mi></msub><mo>:</mo><msup><mi>X</mi> <mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup><mo>→</mo><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\veps_X: X^{***} \to X^*</annotation></semantics></math> is sometimes known as the Dixmier projection from the third dual of a Banach space to (the canonical image of) its first dual; note that this map is weak-star-to-weak-star continuous. (It is a projection in the sense of vector spaces, by the triangle identity for the adjunction.)</p> <h2 id="definition">Definition</h2> <p>A <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a> <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 <strong>reflexive</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>κ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\kappa_X</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bant</mi></mrow><annotation encoding="application/x-tex">Bant</annotation></semantics></math>. If we furthermore grant ourselves Hahn-Banach, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>κ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\kappa_X</annotation></semantics></math> will even be an <a class="existingWikiWord" href="/nlab/show/isometric+isomorphism">isometric isomorphism</a>: an isomorphism in the category of Banach spaces and <a class="existingWikiWord" href="/nlab/show/short+linear+maps">short linear maps</a>.</p> <h2 id="properties">Properties</h2> <p>(…)</p> <p>If two <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>s are isomorphic as TVSes (but not necessarily isometrically isomorphic), then either both are reflexive or both are non-reflexive.</p> <p>There is a nice proof that closed subspaces of reflexive <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>s are reflexive (due to Linton? ) using naturality of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>κ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\kappa_X</annotation></semantics></math>.</p> <p>It turns out that if <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/Banach+space">Banach space</a>, then it is reflexive if and only if its (norm-)closed unit ball is compact in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mi>X</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\sigma(X,X^*)</annotation></semantics></math>-topology. In particular, if <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 reflexive and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E\to X</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">X\to F</annotation></semantics></math> are bounded linear operators, then the composition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">E\to F</annotation></semantics></math> is <span class="newWikiWord">weakly compact<a href="/nlab/new/weakly+compact">?</a></span> as a linear operator.</p> <p>A theorem of Davis-Figiel-Johnson-Pelczynski (1974) tells us that the converse is true: every <span class="newWikiWord">weakly compact<a href="/nlab/new/weakly+compact">?</a></span> linear operator between Banach spaces factors through some reflexive Banach space. The intermediate space is constructed by real interpolation and (at least as usually presented) does not seem to be canonical in any way.</p> <h2 id="examples_and_counterexamples">Examples and counterexamples</h2> <p>By the <span class="newWikiWord">Riesz duality theorem<a href="/nlab/new/Riesz+duality+theorem">?</a></span>, every <a class="existingWikiWord" href="/nlab/show/separable+space">separable</a> <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> is reflexive.</p> <p>The <a class="existingWikiWord" href="/nlab/show/Lebesgue+space">Lebesgue space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>l</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">l^1</annotation></semantics></math> is reflexive in <a class="existingWikiWord" href="/nlab/show/dream+mathematics">dream mathematics</a>, but in <a class="existingWikiWord" href="/nlab/show/classical+mathematics">classical mathematics</a> it is not.</p> <p>In <a class="existingWikiWord" href="/nlab/show/dream+mathematics">dream mathematics</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>l</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">l^\infty</annotation></semantics></math> is reflexive; but its closed subspace <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">c_0</annotation></semantics></math> is not. On the other hand, if one works in a setting where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>κ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\kappa_X</annotation></semantics></math> is a monomorphism, then closed subspaces of reflexive Banach spaces are reflexive (see above)</p> <p>James (1950) constructed a separable non-reflexive Banach space which is isomorphic as a TVS to its second dual; nowadays this is known as the <strong>James space</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>. More is true: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>κ</mi> <mi>J</mi></msub><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">\kappa_J(J)J</annotation></semantics></math> is a codimension-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math> subspace of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>J</mi> <mrow><mo>*</mo><mo>*</mo></mrow></msup></mrow><annotation encoding="application/x-tex">J^{**}</annotation></semantics></math>.</p> <p>One can renorm <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> to obtain a non-reflexive Banach space which is isometrically isomorphic as a Banach space to its second dual (James, 1951).</p> <h2 id="references">References</h2> <ul> <li> <p><a href="http://www.ams.org/mathscinet-getitem?mr=39915">MR0039915</a> (12,616b) James, Robert C. Bases and reflexivity of Banach spaces. <a href="http://dx.doi.org/10.2307/1969430">Ann. of Math. (2) 52, (1950). 518–527.</a></p> </li> <li> <p><a href="http://www.ams.org/mathscinet-getitem?mr=44024">MR0044024</a> (13,356d) James, Robert C. A non-reflexive Banach space isometric with its second conjugate space. Proc. Nat. Acad. Sci. U. S. A. 37, (1951). 174–177.</p> </li> <li> <p><a href="http://www.ams.org/mathscinet-getitem?mr=0355536">MR0355536</a> (50 #8010) Davis, W. J.; Figiel, T.; Johnson, W. B.; Pełczyński, A. Factoring weakly compact operators. J. Functional Analysis 17 (1974), 311–327.</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 23, 2021 at 15:02:16. 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