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class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>, <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a></p> </li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ring">ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-module">(∞,n)-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field">field</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-field">∞-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/2-vector+space">2-vector space</a></p> <p><a class="existingWikiWord" href="/nlab/show/rational+vector+space">rational vector space</a></p> <p><a class="existingWikiWord" href="/nlab/show/real+vector+space">real vector space</a></p> <p><a class="existingWikiWord" href="/nlab/show/complex+vector+space">complex vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+basis">linear basis</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+basis">orthogonal basis</a>, <a class="existingWikiWord" href="/nlab/show/orthonormal+basis">orthonormal basis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a>, <a class="existingWikiWord" href="/nlab/show/antilinear+map">antilinear map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix">matrix</a> (<a class="existingWikiWord" href="/nlab/show/square+matrix">square</a>, <a class="existingWikiWord" href="/nlab/show/invertible+matrix">invertible</a>, <a class="existingWikiWord" href="/nlab/show/diagonal+matrix">diagonal</a>, <a class="existingWikiWord" href="/nlab/show/hermitian+matrix">hermitian</a>, <a class="existingWikiWord" href="/nlab/show/symmetric+matrix">symmetric</a>, …)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a>, <a class="existingWikiWord" href="/nlab/show/matrix+group">matrix group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/eigenspace">eigenspace</a>, <a class="existingWikiWord" href="/nlab/show/eigenvalue">eigenvalue</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inner+product">inner product</a>, <a class="existingWikiWord" href="/nlab/show/Hermitian+form">Hermitian form</a></p> <p><a class="existingWikiWord" href="/nlab/show/Gram-Schmidt+process">Gram-Schmidt process</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <p>(…)</p> </div></div> </div> </div> <h1 id="inner_product_spaces">Inner product spaces</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definitions'>Definitions</a></li> <li><a href='#definite'>Definiteness</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>An <em>inner product</em> on 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>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> (also “scalar product” in the sense of: with values in “<a class="existingWikiWord" href="/nlab/show/scalars">scalars</a>”, namely in 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>𝕂</mi></mrow><annotation encoding="application/x-tex">\mathbb{K}</annotation></semantics></math>) is a pairing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mtext>-</mtext><mo>,</mo><mtext>-</mtext><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle\text{-},\text{-}\rangle</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/vectors">vectors</a> to <a class="existingWikiWord" href="/nlab/show/scalars">scalars</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>v</mi> <mn>2</mn></msub><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>V</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">⟨</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">⟩</mo><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>𝕂</mi></mrow><annotation encoding="application/x-tex"> v_1,\, v_2 \,\in\, V \;\;\;\;\;\;\;\;\; \vdash \;\;\;\;\;\;\;\;\; \langle v_1,\, v_2 \rangle \,\in\, \mathbb{K} </annotation></semantics></math></div> <p>which is <a class="existingWikiWord" href="/nlab/show/bilinear+form">bilinear</a> <em>or rather</em> – namely if <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">\mathbb{K}</annotation></semantics></math> is understood with a <a class="existingWikiWord" href="/nlab/show/star-algebra">star</a>-<a class="existingWikiWord" href="/nlab/show/involution">involution</a> (such as the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> under <a class="existingWikiWord" href="/nlab/show/complex+conjugation">complex conjugation</a>) – <a class="existingWikiWord" href="/nlab/show/sesquilinear+form">sesquilinear</a>, in which case one also speaks of a <em><a class="existingWikiWord" href="/nlab/show/Hermitian+inner+product">Hermitian inner product</a></em>, for definiteness.</p> <p>Often one requires such a pairing to be non-degenerate or even <a class="existingWikiWord" href="/nlab/show/positive-definite">positive-definite</a> in order to qualify as an inner product, standard conventions depend on context.</p> <p>For example, the (<a class="existingWikiWord" href="/nlab/show/Hermitian+inner+product">Hermitian</a>) inner product on a <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> is required to be positive definite, as is that on <a class="existingWikiWord" href="/nlab/show/tangent+spaces">tangent spaces</a> in <a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian geometry</a>, but the inner product on <a class="existingWikiWord" href="/nlab/show/tangent+spaces">tangent spaces</a> in <a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+geometry">pseudo-Riemannian geometry</a> is only required to be non-degenerate.</p> <p>The group of <a class="existingWikiWord" href="/nlab/show/automorphisms">automorphisms</a> of an inner product space is the <a class="existingWikiWord" href="/nlab/show/orthogonal+group+of+an+inner+product+space">orthogonal group of an inner product space</a>.</p> <h2 id="definitions">Definitions</h2> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> over the <a class="existingWikiWord" href="/nlab/show/field">field</a> (or more generally a <a class="existingWikiWord" href="/nlab/show/ring">ring</a>) <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">\mathbb{K}</annotation></semantics></math>. Suppose that <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">\mathbb{K}</annotation></semantics></math> is equipped with an <a class="existingWikiWord" href="/nlab/show/involution">involution</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>↦</mo><mover><mi>r</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">r \mapsto \overline{r}</annotation></semantics></math>, called <em>conjugation</em>; in many examples, this will simply be the <a class="existingWikiWord" href="/nlab/show/identity+function">identity function</a>, but not always (for the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> one typically consider the involution by <a class="existingWikiWord" href="/nlab/show/complex+conjugation">complex conjugation</a>).</p> <p>Then a (<a class="existingWikiWord" href="/nlab/show/Hermitian+inner+product">Hermitian</a>) <em>inner product</em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> 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><mtext>-</mtext><mo>,</mo><mspace width="thinmathspace"></mspace><mtext>-</mtext><mo stretchy="false">⟩</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>V</mi><mo>×</mo><mi>V</mi><mo>→</mo><mi>k</mi></mrow><annotation encoding="application/x-tex"> \langle\text{-},\, \text{-}\rangle \;\colon\; V \times V \to k </annotation></semantics></math></div> <p>that is (1–3) <em>sesquilinear</em> (or <em>bilinear</em> when the involution is the identity) and (4) <em>conjugate-symmetric</em> (or <em>symmetric</em> when the involution is the identity). That is:</p> <ol> <li><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>;</li> <li><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>;</li> <li><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><mo stretchy="false">⟨</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">⟩</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">\langle x, c y \rangle = \langle x, y \rangle c</annotation></semantics></math>;</li> <li><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 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>.</li> </ol> <p>Here we use the <em>physicist's convention</em> that the inner product is antilinear (= conjugate-linear) in the first variable rather than in the second, rather than the <em>mathematician's convention</em>, which is the reverse.</p> <p>Note that we use the same ring as values of the inner product as for <a class="existingWikiWord" href="/nlab/show/scalars">scalars</a>, and that <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><mo stretchy="false">⟨</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">⟩</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">\langle x, c y \rangle = \langle x, y \rangle c</annotation></semantics></math> is written with <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">c</annotation></semantics></math> on the right for the case that we deal with noncommutative division ring.</p> <div class="query"> <p>Are the two conventions really equivalent when <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> is noncommutative? —Toby</p> </div> <p>(The axiom list above is rather redundant. First of all, (1) follows from (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, (1–3) come in pairs, only one of which is needed, since each half follows from the other using (4). It is even possible to derive (3) from (2) under some circumstances.)</p> <p>An <strong>inner product space</strong> is simply a vector space equipped with an inner product.</p> <p>We define a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">‖</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup></mrow><mo lspace="verythinmathspace">:</mo><mi>V</mi><mo>→</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">{\|{-}\|^2}\colon V \to k</annotation></semantics></math> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">‖</mo><mi>x</mi><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup></mrow><mo>=</mo><mo stretchy="false">⟨</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">{\|x\|^2} = \langle x, x \rangle</annotation></semantics></math>; this is called the <strong>norm</strong> 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>. As the notation suggests, it is common to take the norm 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> to be the square root of this expression in contexts where that makes sense, but for us <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">‖</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup></mrow></mrow><annotation encoding="application/x-tex">{\|{-}\|^2}</annotation></semantics></math> is an atomic symbol. The norm 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> is <strong>real</strong> in that it equals its own conjugate, by (4).</p> <h2 id="definite">Definiteness</h2> <p>Notice that, by (1), <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>y</mi><mo stretchy="false">⟩</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\langle 0, y \rangle = 0</annotation></semantics></math> for all <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>. In fact, the <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><mo stretchy="false">{</mo><mi>x</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mo>∀</mo><mi>y</mi><mo>,</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">⟨</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">⟩</mo><mo>=</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{ x \;|\; \forall y,\; \langle x, y \rangle = 0 \}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/linear+subspace">linear subspace</a> of <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>. Of course, we also have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">‖</mo><mn>0</mn><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup></mrow><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">{\|0\|^2} = 0</annotation></semantics></math>, but <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>x</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mrow><mo stretchy="false">‖</mo><mi>x</mi><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup></mrow><mo>=</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{ x \;|\; {\|x\|^2} = 0 \}</annotation></semantics></math> may not be a subspace. These observations motivate some possible conditions on the inner product:</p> <ul> <li>The inner product is <strong>semidefinite</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>x</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mrow><mo stretchy="false">‖</mo><mi>x</mi><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup></mrow><mo>=</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{ x \;|\; {\|x\|^2} = 0 \}</annotation></semantics></math> <em>is</em> closed under addition (and hence is a subspace); it's <strong>indefinite</strong> if there are <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> and <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> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">‖</mo><mi>x</mi><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup></mrow><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">{\|x\|^2} = 0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">‖</mo><mi>y</mi><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup></mrow><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">{\|y\|^2} = 0</annotation></semantics></math> but <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo>+</mo><mi>y</mi><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup></mrow><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">{\|x + y\|^2} \ne 0</annotation></semantics></math>.</li> <li>The inner product is <strong>nondegenerate</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x = 0</annotation></semantics></math> whenever <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 stretchy="false">⟩</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\langle x, y \rangle = 0</annotation></semantics></math> for all <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>; it's <strong>degenerate</strong> if there is <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> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x \ne 0</annotation></semantics></math> but <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 stretchy="false">⟩</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\langle x, y \rangle = 0</annotation></semantics></math> for all <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>.</li> <li>The inner product is <strong>definite</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x = 0</annotation></semantics></math> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">‖</mo><mi>x</mi><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup></mrow><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">{\|x\|^2} = 0</annotation></semantics></math>; there ought to be a term for the condition that there is some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x \ne 0</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">‖</mo><mi>x</mi><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup></mrow><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">{\|x\|^2} = 0</annotation></semantics></math>, so let's call it <strong>nondefinite</strong>.</li> </ul> <p>(In <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a>, we usually want an <a class="existingWikiWord" href="/nlab/show/inequality+relation">inequality relation</a> relative to which the vector-space operations and the inner product are <a class="existingWikiWord" href="/nlab/show/strongly+extensional+function">strongly extensional</a>, to make sense of the conditions with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≠</mo></mrow><annotation encoding="application/x-tex">\ne</annotation></semantics></math> in them. We can also use <a class="existingWikiWord" href="/nlab/show/contrapositives">contrapositives</a> to put <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≠</mo></mrow><annotation encoding="application/x-tex">\ne</annotation></semantics></math> in the other conditions, which makes them stronger if the inequality relation is <a class="existingWikiWord" href="/nlab/show/tight+relation">tight</a>.)</p> <p>An inner product is definite iff it's both semidefinite and nondegenerate. Semidefinite inner products behave very much like definite ones; you can mod out by the elements with norm <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> to get a <a class="existingWikiWord" href="/nlab/show/quotient+vector+space">quotient space</a> with a definite inner product. In a similar way, every inner product space has a nondegenerate quotient.</p> <p>Now suppose that <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> is equipped with a <a class="existingWikiWord" href="/nlab/show/partial+order">partial order</a>. (Note that the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> are standardly so equipped, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>≤</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a \leq b</annotation></semantics></math> iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">b - a</annotation></semantics></math> is a nonnegative real.) Then we can consider other conditions on the inner product:</p> <ul> <li>The inner product is <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><mrow><mo stretchy="false">‖</mo><mi>x</mi><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup></mrow><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">{\|x\|^2} \geq 0</annotation></semantics></math> always.</li> <li>The inner product is <strong>positive definite</strong> if it is both positive and definite, in other words if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">‖</mo><mi>x</mi><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup></mrow><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">{\|x\|^2} \gt 0</annotation></semantics></math> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x \ne 0</annotation></semantics></math>.</li> <li>The inner product is <strong>negative semidefinite</strong>, or simply <strong>negative</strong>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">‖</mo><mi>x</mi><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup></mrow><mo>≤</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">{\|x\|^2} \leq 0</annotation></semantics></math> always.</li> <li>The inner product is <strong>negative definite</strong> if it is both positive and definite, in other words if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">‖</mo><mi>x</mi><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup></mrow><mo><</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">{\|x\|^2} \lt 0</annotation></semantics></math> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x \ne 0</annotation></semantics></math>.</li> </ul> <p>In this case, we have these theorems:</p> <ul> <li>A positive or negative inner product really is semidefinite (as the terminology implies).</li> <li>Conversely, a semidefinite inner product is either positive or negative, at least if we use <a class="existingWikiWord" href="/nlab/show/classical+logic">classical logic</a>. (In <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a>, we can say that a semidefinite inner product is either positive or negative, indeed is positive <a class="existingWikiWord" href="/nlab/show/xor">xor</a> negative, as long as it is nonzero in the sense that at least some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\lang{x,y}\rang</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/apart">apart</a> from zero. And of course, the zero inner product is the only one that is both positive and negative.)</li> <li>Hence, a definite inner product is either positive or negative definite. (We can strengthen this to ‘xor’ and state it constructively if we assume 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 nonzero in that there exists <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> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x \ne 0</annotation></semantics></math>.)</li> <li>Conversely, an inner product is indefinite if and only if some norms are positive and some are negative. (No constructive caveats here!)</li> </ul> <p>Negative (semi)definite inner products behave very much like positive (semi)definite ones; you can turn one into the other by multiplying all inner products by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1</annotation></semantics></math>.</p> <p>The study of positive definite inner product spaces (hence essentially of all semidefinite inner product spaces over partially ordered fields) is essentially the study of <a class="existingWikiWord" href="/nlab/show/Hilbert+spaces">Hilbert spaces</a>. (For Hilbert spaces, one usually uses a <a class="existingWikiWord" href="/nlab/show/topological+field">topological field</a>, typically <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 requires a completeness condition, but this does not effect the algebraic properties much.) The study of indefinite inner product spaces is very different; see the <a href="http://secure.wikimedia.org/wikipedia/en/wiki/Krein_space">English Wikipedia article</a> on <span class="newWikiWord">Krein space<a href="/nlab/new/Krein+space">?</a></span>s for some of it.</p> <p>All of this definiteness terminology may now be applied to an <em><a class="existingWikiWord" href="/nlab/show/linear+operator">operator</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>, since <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 stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">⟨</mo><mrow><mi>x</mi><mo>,</mo><mi>T</mi><mi>y</mi></mrow><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">(x, y) \mapsto \langle{x, T y}\rangle</annotation></semantics></math> is another inner product (on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">dom</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">\dom T</annotation></semantics></math>, if necessary). See <a class="existingWikiWord" href="/nlab/show/positive+operator">positive operator</a>.</p> <h2 id="examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+spaces">Hilbert spaces</a> (over <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>), for example <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>;</p> </li> <li> <p>Finite-dimensional modules over <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">\mathbb{H}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semisimple+Lie+algebras">semisimple Lie algebras</a> with the negative of the <a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a> are positive-definite inner product spaces.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inner+product+abelian+groups">inner product abelian groups</a></p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/self-dual+object">self-dual object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inner+product+of+vector+bundles">inner product of vector bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bra-ket">bra-ket</a></p> </li> </ul> <h2 id="references">References</h2> <p>Original discussion of <a class="existingWikiWord" href="/nlab/show/Hermitian+inner+products">Hermitian inner products</a> in the context of defining <a class="existingWikiWord" href="/nlab/show/Hilbert+spaces">Hilbert spaces</a> (as mathematical foundations of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>):</p> <ul> <li id="vonNeumann30"> <p><a class="existingWikiWord" href="/nlab/show/John+von+Neumann">John von Neumann</a>, p. 64 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>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/John+von+Neumann">John von Neumann</a>:</p> <p>p. 21 in: <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>pp. 38 in: <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>Textbook accounts in the context of <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Richard+V.+Kadison">Richard V. Kadison</a>, <a class="existingWikiWord" href="/nlab/show/John+R.+Ringrose">John R. Ringrose</a>, §2.1 in: <em>Fundamentals of the theory of operator algebras</em> Vol I <em>Elementary Theory</em>, Graduate Studies in Mathematics <strong>15</strong>, AMS (1997) [<a href="https://bookstore.ams.org/gsm-15">ISBN:978-0-8218-0819-1</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bruce+Blackadar">Bruce Blackadar</a>, §I.1.1 in: <em>Operator Algebras – Theory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-Algebras and von Neumann Algebras</em>, Encyclopaedia of Mathematical Sciences <strong>122</strong>, Springer (2006) [<a href="https://doi.org/10.1007/3-540-28517-2">doi:10.1007/3-540-28517-2</a>]</p> </li> </ul> <p>Discussion of plain inner products (without star-involution) in terms of <a class="existingWikiWord" href="/nlab/show/self-dual+objects">self-dual objects</a> in suitable <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric</a> <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a>:</p> <ul> <li id="Selinger12"><a class="existingWikiWord" href="/nlab/show/Peter+Selinger">Peter Selinger</a>, <em>Autonomous categories in which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>≃</mo><msup><mi>A</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">A \simeq A^\ast</annotation></semantics></math></em>, talk at QPL 2012 (<a class="existingWikiWord" href="/nlab/files/SelingerSelfDual.pdf" title="pdf">pdf</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 20, 2024 at 08:08:11. 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