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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='#Idea'>Idea</a></li> <li><a href='#Definition'>Definition</a></li> <ul> <li><a href='#StandardDefinition'>Standard formulation</a></li> <li><a href='#AlternativeDefinitions'>Alternative formulations</a></li> <ul> <li><a href='#AsBanachSpacesAndMetricSpaces'>As Banach spaces and metric spaces</a></li> </ul> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#banach_spaces'>Banach spaces</a></li> <li><a href='#of_lebesgue_squareintegrable_functions_over_a_manifold'>Of Lebesgue square-integrable functions over a manifold</a></li> <li><a href='#of_squareintegrable_halfdensities'>Of square-integrable half-densities</a></li> </ul> <li><a href='#Properties'>Properties</a></li> <ul> <li><a href='#bases'>Bases</a></li> <li><a href='#cauchyschwarz_inequality'>Cauchy–Schwarz inequality</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="Idea">Idea</h2> <blockquote> <p><em>Dr. von Neumann, ich möchte gerne wissen, was ist denn eigentlich ein Hilbertscher Raum ?</em> <sup id="fnref:1"><a href="#fn:1" rel="footnote">1</a></sup></p> </blockquote> <p>A Hilbert space is (see Def. <a class="maruku-ref" href="#HilbertSpace"></a> for details):</p> <ol> <li> <p>a (<a class="existingWikiWord" href="/nlab/show/real+vector+space">real</a> or, usually, <a class="existingWikiWord" href="/nlab/show/complex+vector+space">complex</a>) <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, possibly of infinite <a class="existingWikiWord" href="/nlab/show/dimension+of+a+vector+space">dimension</a>,</p> </li> <li> <p>equipped with a positive definite <a class="existingWikiWord" href="/nlab/show/Hermitian+form">Hermitian</a> <a class="existingWikiWord" href="/nlab/show/inner+product">inner product</a>,</p> </li> <li> <p>which, as a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, is <a class="existingWikiWord" href="/nlab/show/complete+metric+space">complete</a> with respect to the induced <a class="existingWikiWord" href="/nlab/show/metric">metric</a>.</p> </li> </ol> <p>Hilbert spaces are central to <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a> and specifically to <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>, where they serve as <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">spaces of</a> <a class="existingWikiWord" href="/nlab/show/pure+quantum+states">pure quantum states</a>. Here the <a class="existingWikiWord" href="/nlab/show/inner+product">inner product</a> encodes the <a class="existingWikiWord" href="/nlab/show/probability+amplitudes">probability amplitudes</a> for one <a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a> to “<a class="existingWikiWord" href="/nlab/show/collapse+of+the+wavefunction">collapse</a>” to another one under <a class="existingWikiWord" href="/nlab/show/quantum+measurement">measurement</a>. When the space of pure states is of <a class="existingWikiWord" href="/nlab/show/finite-dimensional+vector+space">finite dimension</a> (as is the case of interest in <a class="existingWikiWord" href="/nlab/show/quantum+information+theory">quantum information theory</a>/<a class="existingWikiWord" href="/nlab/show/quantum+computation">quantum computation</a>) then the completeness condition on a Hilbert space is automatic (see Rem. <a class="maruku-ref" href="#FiniteDimensionalInnerProductSpaces"></a> below), otherwise it naturally encodes the possibility of an infinite number of <a class="existingWikiWord" href="/nlab/show/quantum+measurement">measurement</a> outcomes.</p> <p>Hilbert spaces with (<a class="existingWikiWord" href="/nlab/show/bounded+linear+map">bounded</a>) <a class="existingWikiWord" href="/nlab/show/linear+maps">linear maps</a> between them form a (<a class="existingWikiWord" href="/nlab/show/dagger+category">dagger</a>-)<a class="existingWikiWord" href="/nlab/show/category">category</a>, often denoted <em><a class="existingWikiWord" href="/nlab/show/Hilb">Hilb</a></em> or similar, with the <a class="existingWikiWord" href="/nlab/show/dagger-category">dagger</a>-structure given by sending <a class="existingWikiWord" href="/nlab/show/bounded+linear+maps">bounded linear maps</a> to their <a class="existingWikiWord" href="/nlab/show/adjoint+operators">adjoint operators</a> with respect to the Hermitian inner product. Finite-dimensional Hilbert spaces form a <a class="existingWikiWord" href="/nlab/show/dagger-compact+category">dagger-compact category</a>.</p> <p>See also:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/an+elementary+treatment+of+Hilbert+spaces">an elementary treatment of Hilbert spaces</a>.</li> </ul> <h2 id="Definition">Definition</h2> <p>We first state the</p> <ul> <li><a href="#StandardDefinition">standard formulation</a></li> </ul> <p>of the definition of Hilbert spaces and then various equivalent</p> <ul> <li><a href="#AlternativeDefinitions">alternative formulations</a></li> </ul> <h3 id="StandardDefinition">Standard formulation</h3> <p>We first state the definition as such and then recall the ingredients that go into it:</p> <p> <div class='num_defn' id='HilbertSpace'> <h6>Definition</h6> <p><strong>(Hilbert space)</strong> <br /></p> <p>A <em>Hilbert space</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mi class="mathscript">ℋ</mi><mo>,</mo><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\big(\mathscr{H}, \langle -,-\rangle\big)</annotation></semantics></math> is</p> <ol> <li> <p>a (<a class="existingWikiWord" href="/nlab/show/complex+vector+space">complex</a> or <a class="existingWikiWord" href="/nlab/show/real+vector+space">real</a>) <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi class="mathscript">ℋ</mi></mrow><annotation encoding="application/x-tex">\mathscr{H}</annotation></semantics></math> (possibly infinite-<a class="existingWikiWord" href="/nlab/show/dimension+of+a+vector+space">dimensional</a>),</p> </li> <li> <p>equipped with a <a class="existingWikiWord" href="/nlab/show/Hermitian+form">Hermitian</a> <a class="existingWikiWord" href="/nlab/show/inner+product">inner product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle -,- \rangle</annotation></semantics></math> (Def. <a class="maruku-ref" href="#HermitianInnerProduct"></a>)</p> </li> </ol> <p>which is</p> <ol> <li> <p>complete (Def. <a class="maruku-ref" href="#CompleteInnerProduct"></a>);</p> </li> <li> <p>positive definite (Def. <a class="maruku-ref" href="#DefiniteInnerProduct"></a>).</p> </li> </ol> <p></p> </div> </p> <p> <div class='num_remark' id='MorphismOfHilbertSpaces'> <h6>Remark</h6> <p><strong>(morphisms of Hilbert spaces)</strong> <br /> There are different notions of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> between <a class="existingWikiWord" href="/nlab/show/Hilbert+spaces">Hilbert spaces</a>:</p> <p>At the very least, a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> of <a class="existingWikiWord" href="/nlab/show/Hilbert+spaces">Hilbert spaces</a> is a <a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a> between the underlying <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a>, and this is the default notion usually considered. Notice that such morphisms do not need to respect the <a class="existingWikiWord" href="/nlab/show/inner+product">inner product</a>-<a class="existingWikiWord" href="/nlab/show/structure">structure</a>. Often one restricts to <a class="existingWikiWord" href="/nlab/show/bounded+linear+maps">bounded linear maps</a> or sometimes <a class="existingWikiWord" href="/nlab/show/short+maps">short maps</a> (see at <em><a href="Banach+space#morphisms">maps of Banach spaces</a></em>) which respect part of the inner product structure.</p> <p>Instead of being fully respected by maps, the inner product on Hilbert spaces induces the <a class="existingWikiWord" href="/nlab/show/dagger-category">dagger</a> <a class="existingWikiWord" href="/nlab/show/involution">involution</a> on their <a class="existingWikiWord" href="/nlab/show/hom-spaces">hom-spaces</a> given by sending any <a class="existingWikiWord" href="/nlab/show/bounded+linear+map">bounded linear map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo lspace="verythinmathspace">:</mo><msub><mi class="mathscript">ℋ</mi> <mn>1</mn></msub><mo>→</mo><msub><mi class="mathscript">ℋ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">A \colon \mathscr{H}_1 \to \mathscr{H}_2</annotation></semantics></math> to its <a class="existingWikiWord" href="/nlab/show/Hermitian+form">Hermitian</a> <a class="existingWikiWord" href="/nlab/show/adjoint+operator">adjoint operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mo>†</mo></msup><mo lspace="verythinmathspace">:</mo><msub><mi class="mathscript">ℋ</mi> <mn>2</mn></msub><mo>→</mo><msub><mi class="mathscript">ℋ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">A^\dagger \colon \mathscr{H}_2 \to \mathscr{H}_1</annotation></semantics></math>, characterized by</p> <div class="maruku-equation" id="eq:AdjointnessRelation"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><msup><mi>A</mi> <mo>†</mo></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">⟩</mo> <mn>1</mn></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>A</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">⟩</mo> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> \langle A^\dagger(-), -\rangle_{1} \;=\; \langle -, A(-)\rangle_{2} </annotation></semantics></math></div> <p></p> </div> </p> <p> <div class='num_remark' id='FiniteDimensionalInnerProductSpaces'> <h6>Remark</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/finite-dimensional+Hilbert+space">finite-dimensional Hilbert space</a>)</strong> <br /> In the case that the underlying vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi class="mathscript">ℋ</mi></mrow><annotation encoding="application/x-tex">\mathscr{H}</annotation></semantics></math> happens to be <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+space">finite dimensional vector space</a>, any hermitian inner product is necessarily complete. Therefore:</p> <p>A <em>finite-dimensional Hilbert space</em> is equivalently a positive-definite <a class="existingWikiWord" href="/nlab/show/hermitian+inner+product">hermitian inner product</a>-space of <a class="existingWikiWord" href="/nlab/show/finite+set">finite</a> <a class="existingWikiWord" href="/nlab/show/dimension+of+a+vector+space">dimension</a>.</p> <p>Finite-dimensional Hilbert spaces form a <a class="existingWikiWord" href="/nlab/show/dagger-compact+category">dagger-compact category</a> and play a central role in <a class="existingWikiWord" href="/nlab/show/quantum+information+theory">quantum information theory</a> and <a class="existingWikiWord" href="/nlab/show/quantum+computation">quantum computation</a>, see also at <em><a class="existingWikiWord" href="/nlab/show/quantum+information+theory+via+dagger-compact+categories">quantum information theory via dagger-compact categories</a></em>.</p> </div> </p> <p><br /></p> <p>We recall now the meaning of the concepts entering Def. <a class="maruku-ref" href="#HilbertSpace"></a>. In the following, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> over the <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>∈</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">z \in \mathbb{C}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/complex+number">complex number</a>, we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>z</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{z}</annotation></semantics></math> for its <a class="existingWikiWord" href="/nlab/show/complex+conjugate">complex conjugate</a>.</p> <p> <div class='num_remark' id='AlternativeGroundFields'> <h6>Remark</h6> <p><strong>(alternative ground fields)</strong> <br /> The following applies verbatim also for the ground field of <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> <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>, in which case the sesquilinear inner product <a href="#HermitianInnerProduct">below</a> becomes <a class="existingWikiWord" href="/nlab/show/bilinear+form">bilinear</a> and one speaks of <em>real Hilbert spaces</em>. Under mild assumptions, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> and <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> are the only possible <a class="existingWikiWord" href="/nlab/show/ground+fields">ground fields</a> for Hilbert spaces (see <a href="https://math.stackexchange.com/a/4184099/58526">MO:a/4184099</a>).</p> </div> </p> <p> <div class='num_defn' id='HermitianInnerProduct'> <h6>Definition</h6> <p><strong>(Hermitian inner product)</strong> <br /> A <em><a class="existingWikiWord" href="/nlab/show/Hermitian+form">Hermitian</a> <a class="existingWikiWord" href="/nlab/show/inner+product">inner product</a></em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi class="mathscript">ℋ</mi></mrow><annotation encoding="application/x-tex">\mathscr{H}</annotation></semantics></math> is a function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi class="mathscript">ℋ</mi><mo>×</mo><mi class="mathscript">ℋ</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex"> \langle {-},{-} \rangle \;\colon\; \mathscr{H} \times \mathscr{H} \to \mathbb{C} </annotation></semantics></math></div> <p>that is</p> <ol> <li> <p><strong>sesquilinear</strong>:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo stretchy="false">⟩</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> \langle 0, x \rangle = 0 </annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo stretchy="false">⟩</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> \langle x, 0 \rangle = 0 </annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">⟩</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo stretchy="false">⟩</mo><mo>+</mo><mo stretchy="false">⟨</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex"> \langle x + y, z \rangle = \langle x, z \rangle + \langle y, z \rangle </annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo stretchy="false">⟩</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">⟩</mo><mo>+</mo><mo stretchy="false">⟨</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex"> \langle x, y + z \rangle = \langle x, y \rangle + \langle x, z \rangle </annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>c</mi><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">⟩</mo><mo>=</mo><mover><mi>c</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">⟨</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex"> \langle c x, y \rangle = \bar{c} \langle x, y \rangle </annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>x</mi><mo>,</mo><mi>c</mi><mi>y</mi><mo stretchy="false">⟩</mo><mo>=</mo><mi>c</mi><mo stretchy="false">⟨</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex"> \langle x, c y \rangle = c \langle x, y \rangle </annotation></semantics></math>;</p> </li> </ol> </li> <li> <p><strong>conjugate-symmetric</strong>:</p> <div class="maruku-equation" id="eq:ConjugateSymmetryOfInnerProduct"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">⟩</mo><mo>=</mo><mover><mrow><mo stretchy="false">⟨</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo stretchy="false">⟩</mo></mrow><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex"> \langle x, y \rangle = \overline{\langle y, x \rangle} </annotation></semantics></math></div></li> </ol> <p></p> </div> </p> <p> <div class='num_remark' id='ConventionsForInnerProduct'> <h6>Remark</h6> <p><strong>(convention for the inner product)</strong> <br /> Def. <a class="maruku-ref" href="#HermitianInnerProduct"></a> uses the <em>physicist's convention</em> that the inner product is conjugate-linear in the first variable rather than in the second, instead of the <em>mathematician's convention</em>, which is the reverse. The physicist's convention fits in a little better with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/2-Hilbert+space">Hilbert spaces</a>.</p> </div> </p> <p> <div class='num_remark'> <h6>Remark</h6> <p>The axiom list in Def. <a class="maruku-ref" href="#HermitianInnerProduct"></a> is rather redundant. First of all, (1.1) follows from (1.3) by setting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">c = 0</annotation></semantics></math>; besides that, (1.1–1.3) come in pairs, only one of which is needed, since each half follows from the other using (2). It is even possible to derive (1.3) from (1.2) by supposing that <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 <a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a> and that the inner product is <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous</a> (which, as we will see, is always true anyway for a Hilbert space).</p> </div> </p> <p> <div class='num_defn' id='DefiniteInnerProduct'> <h6>Definition</h6> <p><strong>(definite inner product)</strong> <br /> Given an inner product according to Def. <a class="maruku-ref" href="#HermitianInnerProduct"></a>, consider the (norm square) function</p> <div class="maruku-equation" id="eq:NormSquare"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">‖</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace></mrow></mpadded><mi class="mathscript">ℋ</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>ℝ</mi><mo>↪</mo><mi>ℂ</mi></mtd></mtr> <mtr><mtd><mi>v</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mo stretchy="false">⟨</mo><mi>v</mi><mo>,</mo><mi>v</mi><mo stretchy="false">⟩</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathllap{ \|{-}\|^2 \;\colon\; } \mathscr{H} &\longrightarrow& \mathbb{R} \hookrightarrow \mathbb{C} \\ v &\mapsto& \langle v, v \rangle } </annotation></semantics></math></div> <p>Notice that this takes only real values, by <a class="maruku-eqref" href="#eq:ConjugateSymmetryOfInnerProduct">(2)</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/Hermitian+form">Hermitian</a> <a class="existingWikiWord" href="/nlab/show/inner+product">inner product</a> is called:</p> <ul> <li> <p><strong>positive semidefinite</strong>, or simply <strong>positive</strong>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">‖</mo><mi>v</mi><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\|v\|^2 \geq 0</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><mi class="mathscript">ℋ</mi></mrow><annotation encoding="application/x-tex">v \in \mathscr{H}</annotation></semantics></math>;</p> </li> <li> <p><strong>non-degenerate</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">‖</mo><mi>v</mi><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\|v\|^2 = 0</annotation></semantics></math> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">v = 0</annotation></semantics></math>;</p> </li> <li> <p><strong>positive definite</strong> if it is both positive and non-degenerate.</p> </li> </ul> <p></p> </div> </p> <p> <div class='num_remark'> <h6>Remark</h6> <p>An inner product is called <em>indefinite</em> if some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">‖</mo><mi>v</mi><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\|v\|^2</annotation></semantics></math> are positive and some are negative.</p> </div> </p> <p> <div class='num_defn' id='CompleteInnerProduct'> <h6>Definition</h6> <p><strong>(complete inner product)</strong> <br /> The inner product (Def. <a class="maruku-ref" href="#HermitianInnerProduct"></a>) is <strong>complete</strong> if, given any infinite <a class="existingWikiWord" href="/nlab/show/sequence">sequence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><mi>…</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(v_1, v_2, \ldots)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi class="mathscript">𝒱</mi></mrow><annotation encoding="application/x-tex">\mathscr{V}</annotation></semantics></math> such that we have the <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit</a></p> <div class="maruku-equation" id="eq:Cauchy"><span class="maruku-eq-number">(4)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi> <mrow><mi>m</mi><mo>,</mo><mi>n</mi><mo>→</mo><mn>∞</mn></mrow></munder><msup><mrow><mo>‖</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mi>m</mi></mrow> <mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></munderover><msub><mi>v</mi> <mi>i</mi></msub><mo>‖</mo></mrow> <mn>2</mn></msup><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \lim_{m,n\to\infty} \left\|\sum_{i=m}^{m+n} v_i\right\|^2 = 0 \,, </annotation></semantics></math></div> <p>then there exists a (necessarily unique) <em><a class="existingWikiWord" href="/nlab/show/sum">sum</a></em> <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> such that</p> <div class="maruku-equation" id="eq:converge"><span class="maruku-eq-number">(5)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi> <mrow><mi>n</mi><mo>→</mo><mn>∞</mn></mrow></munder><msup><mrow><mo>‖</mo><mi>S</mi><mo>−</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></munderover><msub><mi>v</mi> <mi>i</mi></msub><mo>‖</mo></mrow> <mn>2</mn></msup><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \lim_{n\to\infty} \left\|S - \sum_{i=1}^n v_i\right\|^2 = 0 \,. </annotation></semantics></math></div> <p>If the inner product is definite (Def. <a class="maruku-ref" href="#DefiniteInnerProduct"></a>), then this sum, if it exists, must be unique, and we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mn>∞</mn></munderover><msub><mi>v</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex"> S = \sum_{i=1}^\infty v_i </annotation></semantics></math></div> <p>(with the right-hand side undefined if no such sum exists).</p> <p></p> </div> </p> <h3 id="AlternativeDefinitions">Alternative formulations</h3> <h4 id="AsBanachSpacesAndMetricSpaces">As Banach spaces and metric spaces</h4> <p>If an inner product (Def. <a class="maruku-ref" href="#HermitianInnerProduct"></a>) is positive (Def. <a class="maruku-ref" href="#DefiniteInnerProduct"></a>), then there exists the principal <a class="existingWikiWord" href="/nlab/show/square+root">square root</a> of the norm square <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">‖</mo><mi>v</mi><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup><mo>=</mo><mo stretchy="false">⟨</mo><mi>v</mi><mo>,</mo><mi>v</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\|v\|^2 = \langle v, v \rangle</annotation></semantics></math> <a class="maruku-eqref" href="#eq:NormSquare">(3)</a> being the <em><a class="existingWikiWord" href="/nlab/show/norm">norm</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">‖</mo><mi>v</mi><mo stretchy="false">‖</mo></mrow><annotation encoding="application/x-tex">\|v\|</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><mi class="mathscript">ℋ</mi></mrow><annotation encoding="application/x-tex">v \in \mathscr{H}</annotation></semantics></math>.</p> <p>This norm satisfies all of the requirements of a <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>. It additionally satisfies the <em>parallelogram law</em></p> <div class="maruku-equation" id="eq:ParallelogramLaw"><span class="maruku-eq-number">(6)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">‖</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>v</mi> <mn>2</mn></msub><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup><mo>+</mo><mo stretchy="false">‖</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>−</mo><msub><mi>v</mi> <mn>2</mn></msub><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mn>2</mn><mo stretchy="false">‖</mo><msub><mi>v</mi> <mn>1</mn></msub><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup><mo>+</mo><mn>2</mn><mo stretchy="false">‖</mo><msub><mi>v</mi> <mn>2</mn></msub><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \|v_1 + v_2\|^2 + \|v_1 - v_2\|^2 \;=\; 2 \|v_1\|^2 + 2 \|v_2\|^2 \,. </annotation></semantics></math></div> <p>which not all Banach spaces need satisfy. (The name of this law comes from its geometric interpretation: the norms in the left-hand side are the lengths of the diagonals of a parallelogram, while the norms in the right-hand side are the lengths of the sides.)</p> <p>Furthermore, any Banach space satsifying the parallelogram law <a class="maruku-eqref" href="#eq:ParallelogramLaw">(6)</a> has a unique inner product that reproduces the norm, defined by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">⟩</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mfrac><mn>1</mn><mn>4</mn></mfrac><mrow><mo>(</mo><mo stretchy="false">‖</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>v</mi> <mn>2</mn></msub><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup><mo>−</mo><mo stretchy="false">‖</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>−</mo><msub><mi>v</mi> <mn>2</mn></msub><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup><mo>−</mo><mi mathvariant="normal">i</mi><mo stretchy="false">‖</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>+</mo><mi mathvariant="normal">i</mi><msub><mi>v</mi> <mn>2</mn></msub><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup><mo>+</mo><mi mathvariant="normal">i</mi><mo stretchy="false">‖</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>−</mo><mi mathvariant="normal">i</mi><msub><mi>v</mi> <mn>2</mn></msub><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup><mo>)</mo></mrow><mo>,</mo></mrow><annotation encoding="application/x-tex"> \langle v_1, v_2 \rangle \;\coloneqq\; \frac{1}{4} \left( \|v_1 + v_2\|^2 - \|v_1 - v_2\|^2 - \mathrm{i} \|v_1 + \mathrm{i}v_2\|^2 + \mathrm{i} \|v_1 - \mathrm{i}v_2\|^2 \right) , </annotation></semantics></math></div> <p>or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mo stretchy="false">‖</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>v</mi> <mn>2</mn></msub><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup><mo>−</mo><mo stretchy="false">‖</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>−</mo><msub><mi>v</mi> <mn>2</mn></msub><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\frac{1}{2}(\|v_1 + v_2\|^2 - \|v_1 - v_2\|^2)</annotation></semantics></math> in the real case.</p> <p>Therefore: A Hilbert space is equivalently <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a> that satisfies the parallelogram law <a class="maruku-eqref" href="#eq:ParallelogramLaw">(6)</a>.</p> <p>This actually works a bit more generally; a positive semidefinite inner product space is a pseudonormed vector space that satisfies the parallelogram law. (We cannot, however, recover an indefinite inner product from a norm.)</p> <p>Moreover, in any positive semidefinite inner product space, let the <strong>distance</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d(x,y)</annotation></semantics></math> be</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">‖</mo><mi>y</mi><mo>−</mo><mi>x</mi><mo stretchy="false">‖</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d(x,y) \;\coloneqq\; \|y - x\| \,. </annotation></semantics></math></div> <p>Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/pseudometric">pseudometric</a>; and it is a complete metric if and only if we have a Hilbert space.</p> <p>In fact, the axioms of a <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a> (or pseudonormed vector space) can be written entirely in terms of the metric; we can also state the parallelogram law as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>+</mo><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>y</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>=</mo><mn>2</mn><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mn>0</mn><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>+</mo><mi>y</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>.</mo></mrow><annotation encoding="application/x-tex"> d(x,y)^2 + d(x,-y)^2 = 2 d(x,0)^2 + 2 d(x,x+y)^2 .</annotation></semantics></math></div> <p>In definitions, it is probably most common to see the metric introduced only to state the completeness requirement. Indeed, <a class="maruku-eqref" href="#eq:Cauchy">(4)</a> says that the sequence of partial sums is a <a class="existingWikiWord" href="/nlab/show/Cauchy+sequence">Cauchy sequence</a>, while <a class="maruku-eqref" href="#eq:converge">(5)</a> says that the sequence of partial sums converges to <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>.</p> <h2 id="examples">Examples</h2> <h3 id="banach_spaces">Banach spaces</h3> <p>All of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-parametrised examples at <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a> apply if you take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">p = 2</annotation></semantics></math>.</p> <p>In particular, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℂ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}^n</annotation></semantics></math> is a complex Hilbert space with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">⟩</mo><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>u</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></munderover><msub><mover><mi>x</mi><mo stretchy="false">¯</mo></mover> <mi>u</mi></msub><msub><mi>y</mi> <mi>u</mi></msub><mo>.</mo></mrow><annotation encoding="application/x-tex"> \langle x, y \rangle = \sum_{u=1}^n \bar{x}_u y_u .</annotation></semantics></math></div> <p>Any subfield <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> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> gives a positive definite inner product space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>K</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">K^n</annotation></semantics></math> whose completion is either <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℂ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}^n</annotation></semantics></math>. In particular, the <a class="existingWikiWord" href="/nlab/show/cartesian+space">cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> is a real Hilbert space; the geometric notions of distance and angle defined above agree with ordinary <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean geometry</a> for this example.</p> <h3 id="of_lebesgue_squareintegrable_functions_over_a_manifold">Of Lebesgue square-integrable functions over a manifold</h3> <p>The <a class="existingWikiWord" href="/nlab/show/L-2-spaces">L-</a> Hilbert spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L^2(\mathbb{R})</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L^2([0,1])</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L^2(\mathbb{R}^3)</annotation></semantics></math>, etc (real or complex) are very well known. In general, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L^2(X)</annotation></semantics></math> for <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 <a class="existingWikiWord" href="/nlab/show/measure+space">measure space</a> consists of the almost-everywhere defined functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> from <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 the scalar field (<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> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math>) such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∫</mo><mo stretchy="false">|</mo><mi>f</mi><msup><mo stretchy="false">|</mo> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex"> \int |f|^2 </annotation></semantics></math> converges to a finite number, with functions identified if they are equal almost everywhere; we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">⟩</mo><mo>=</mo><mo>∫</mo><mover><mi>f</mi><mo stretchy="false">¯</mo></mover><mi>g</mi></mrow><annotation encoding="application/x-tex">\langle f, g\rangle = \int \bar{f} g</annotation></semantics></math>, which converges by the Cauchy–Schwarz inequality. In the specific cases listed (and in general, when <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/locally+compact+space">locally compact</a> <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>), we can also get this space by completing the positive definite inner product space of compactly supported continuous functions.</p> <h3 id="of_squareintegrable_halfdensities">Of square-integrable half-densities</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/canonical+Hilbert+space+of+half-densities">canonical Hilbert space of half-densities</a></li> </ul> <h2 id="Properties">Properties</h2> <h3 id="bases">Bases</h3> <p>A basic result is that abstractly, Hilbert spaces of the same dimension are all of the same type: every Hilbert space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> admits an <a class="existingWikiWord" href="/nlab/show/orthonormal+basis">orthonormal basis</a>, meaning a <a class="existingWikiWord" href="/nlab/show/subset">subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊆</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">S \subseteq H</annotation></semantics></math> whose inclusion map extends (necessarily uniquely) to an isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>l</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>→</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">l^2(S) \to H</annotation></semantics></math></div> <p>of Hilbert spaces. Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>l</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">l^2(S)</annotation></semantics></math> is the vector space consisting of those <a class="existingWikiWord" href="/nlab/show/function">function</a>s <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> from <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> to the scalar field such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">‖</mo><mi>x</mi><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>u</mi><mo>:</mo><mi>S</mi></mrow></munder><mo stretchy="false">|</mo><msub><mi>x</mi> <mi>u</mi></msub><msup><mo stretchy="false">|</mo> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex"> \|x\|^2 = \sum_{u: S} |x_u|^2 </annotation></semantics></math></div> <p>converges to a finite number; this may also be obtained by completing the vector space of formal linear combinations of elements of <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> with an inner product uniquely determined by the rule</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">⟩</mo><mo>=</mo><msub><mi>δ</mi> <mrow><mi>u</mi><mi>v</mi></mrow></msub><mspace width="2em"></mspace><mi>u</mi><mo>,</mo><mi>v</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">\langle u, v \rangle = \delta_{u v} \qquad u, v \in S</annotation></semantics></math></div> <p>in which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mrow><mi>u</mi><mi>v</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\delta_{u v}</annotation></semantics></math> denotes <a class="existingWikiWord" href="/nlab/show/Kronecker+delta">Kronecker delta</a>. We thus have, in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>l</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">l^2(S)</annotation></semantics></math>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">⟩</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>u</mi><mo>:</mo><mi>S</mi></mrow></munder><msub><mover><mi>x</mi><mo stretchy="false">¯</mo></mover> <mi>u</mi></msub><msub><mi>y</mi> <mi>u</mi></msub><mo>.</mo></mrow><annotation encoding="application/x-tex"> \langle x, y \rangle = \sum_{u: S} \bar{x}_u y_u .</annotation></semantics></math></div> <p>(This sum converges by the Cauchy–Schwarz inequality.)</p> <p>In general, this result uses the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> (usually in the form of <a class="existingWikiWord" href="/nlab/show/Zorn%27s+lemma">Zorn's lemma</a> and <a class="existingWikiWord" href="/nlab/show/excluded+middle">excluded middle</a>) in its proof, and is equivalent to it. However, the result for <a class="existingWikiWord" href="/nlab/show/separable+space">separable</a> Hilbert spaces needs only <a class="existingWikiWord" href="/nlab/show/dependent+choice">dependent choice</a> and so is <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive</a> by most schools' standards. Even without dependent choice, explicit orthornormal bases for particular <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L^2(X)</annotation></semantics></math> can often be produced using <a class="existingWikiWord" href="/nlab/show/approximation+of+the+identity">approximation of the identity</a> techniques, often in concert with a <a class="existingWikiWord" href="/nlab/show/Gram-Schmidt+process">Gram-Schmidt process</a>.</p> <p>In particular, all infinite-dimensional separable Hilbert spaces are abstractly isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>l</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>ℕ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">l^2(\mathbb{N})</annotation></semantics></math>.</p> <h3 id="cauchyschwarz_inequality">Cauchy–Schwarz inequality</h3> <p>The <em>Schwarz inequality</em> (or <em>Cauchy–Буняковский–Schwarz inequality</em>, etc) is very handy:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mo stretchy="false">⟨</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">⟩</mo><mo stretchy="false">|</mo><mo>≤</mo><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo><mo stretchy="false">‖</mo><mi>y</mi><mo stretchy="false">‖</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> |\langle x, y \rangle| \leq \|x\| \|y\| .</annotation></semantics></math></div> <p>This is really two theorems (at least): an abstract theorem that the inequality holds in any Hilbert space, and concrete theorems that it holds when the inner product and norm are defined by the formulas used in the examples <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L^2(X)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>l</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">l^2(S)</annotation></semantics></math> above. The concrete theorems apply even to functions that don't belong to the Hilbert space and so prove that the inner product converges whenever the norms converge. (A somewhat stronger result is needed to conclude this convergence constructively; it may be found in Errett Bishop's book.)</p> <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/rigged+Hilbert+space">rigged Hilbert space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+C-star-module">Hilbert C-star-module</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+bimodule">Hilbert bimodule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measurable+field+of+Hilbert+spaces">measurable field of Hilbert spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+vector+space">Kähler vector space</a></p> </li> </ul> <h2 id="references">References</h2> <p>Hilbert spaces were effectively introduced and used by <a class="existingWikiWord" href="/nlab/show/David+Hilbert">David Hilbert</a> and others in the context of <a class="existingWikiWord" href="/nlab/show/integration">integration</a> theory, but the terminology and the formal definition is due to:</p> <ul> <li id="vonNeumann30"><a class="existingWikiWord" href="/nlab/show/John+von+Neumann">John von Neumann</a>, §I in: <em>Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren</em>, Math. Ann. <strong>102</strong> (1930) 49–131 [<a href="https://doi.org/10.1007/BF01782338">doi:10.1007/BF01782338</a>]</li> </ul> <p>motivated from laying foundations for <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>:</p> <ul> <li id="vonNeumann32"> <p><a class="existingWikiWord" href="/nlab/show/John+von+Neumann">John von Neumann</a>:</p> <p><em>Mathematische Grundlagen der Quantenmechanik</em>, Springer (1932, 1971) [<a href="https://link.springer.com/book/10.1007/978-3-642-96048-2">doi:10.1007/978-3-642-96048-2</a>]</p> <p><em>Mathematical Foundations of Quantum Mechanics</em> Princeton University Press (1955) [<a href="https://doi.org/10.1515/9781400889921">doi:10.1515/9781400889921</a>, <a href="https://en.wikipedia.org/wiki/Mathematical_Foundations_of_Quantum_Mechanics">Wikipedia entry</a>]</p> </li> </ul> <p>(where the definition occupies section II.1)</p> <p>but see:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Miklos+R%C3%A9dei">Miklos Rédei</a>, <em>Why John von Neumann did not Like the Hilbert Space formalism of quantum mechanics (and what he liked instead)</em>, Studies in History and Philosophy of Modern Physics <strong>27</strong> 4 (1996) 493-510 [<a href="https://doi.org/10.1016/S1355-2198(96)00017-2">doi:10.1016/S1355-2198(96)00017-2</a>]</li> </ul> <p>Early history:</p> <ul> <li id="MacLane88"><a class="existingWikiWord" href="/nlab/show/Saunders+Mac+Lane">Saunders Mac Lane</a>: <em>Hilbert space</em>, §5 in: <em>Concepts and Categories in Perspective</em>, in: P. Duren, <em>A century of mathematics in America</em> Part 1, AMS (1988) 323-365. [<a href="http://www.ams.org/samplings/math-history/hmath1-maclane25.pdf">pdf</a>, <a href="https://www.ams.org/publicoutreach/math-history/hmath1-index">ISBN:0-8218-0124-4</a>]</li> </ul> <p>Textbook account:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Nicolas+Bourbaki">Nicolas Bourbaki</a>, §V in: <em>Topological Vector Spaces</em>, Chapters 1–5, Masson (1981), Springer (2003) [<a href="https://doi.org/10.1007/978-3-642-61715-7">doi:10.1007/978-3-642-61715-7</a>]</li> </ul> <p>Review:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/George+Mackey">George Mackey</a>, <em>The Mathematical Foundations of Quamtum Mechanics</em> A Lecture-note Volume, The mathematical physics monograph series. Princeton university (1963)</p> </li> <li> <p>E. Prugoveĉki, <em>Quantum mechanics in Hilbert Space</em>. Academic Press (1971)</p> </li> </ul> <p>An <a class="existingWikiWord" href="/nlab/show/axiom">axiomatic</a> characterization of the <a class="existingWikiWord" href="/nlab/show/dagger-category">dagger-category</a> <a class="existingWikiWord" href="/nlab/show/Hilb">Hilb</a> of Hilbert spaces, with <a class="existingWikiWord" href="/nlab/show/linear+maps">linear maps</a> between them:</p> <ul> <li id="HeunenKornell21"> <p><a class="existingWikiWord" href="/nlab/show/Chris+Heunen">Chris Heunen</a>, <a class="existingWikiWord" href="/nlab/show/Andre+Kornell">Andre Kornell</a>, <em>Axioms for the category of Hilbert spaces</em>, PNAS <strong>119</strong> 9 (2022) e2117024119 [<a href="https://arxiv.org/abs/2109.07418">arXiv:2109.07418</a>, <a href="https://doi.org/10.1073/pnas.2117024119">doi:10.1073/pnas.2117024119</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chris+Heunen">Chris Heunen</a>, <a class="existingWikiWord" href="/nlab/show/Andre+Kornell">Andre Kornell</a>, <a class="existingWikiWord" href="/nlab/show/Nesta+van+der+Schaaf">Nesta van der Schaaf</a>, <em>Axioms for the category of Hilbert spaces and linear contractions</em> [<a href="https://arxiv.org/abs/2211.02688">arXiv:2211.02688</a>]</p> </li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/analysis">analysis</a></div><div class="footnotes"><hr /><ol><li id="fn:1"> <p><em>Dr. von Neumann, I would like to know what is a Hilbert space?</em> – Question asked by <a class="existingWikiWord" href="/nlab/show/David+Hilbert">David Hilbert</a>, in a 1929 talk by <a class="existingWikiWord" href="/nlab/show/John+von+Neumann">John von Neumann</a> in Göttingen (cf. <a href="#vonNeumann30">von Neumann 1930, §I</a>). The anecdote is narrated for instance in <a href="#MacLane88">MacLane 1988, §5</a>. (We have corrected ‘dann’ in the original quotation to the more likely ‘denn’, in either case expressing a certain sense of puzzlement that is not quite captured by the direct English translation.) <a href="#fnref:1" rev="footnote">↩</a></p> </li></ol></div></body></html> </div> <div class="revisedby"> <p> Last revised on November 16, 2023 at 07:32:09. 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