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linear map in nLab
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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="linear_algebra">Linear algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/higher+linear+algebra">higher linear algebra</a></strong></p> <h2 id="ingredients">Ingredients</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>, <a 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="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#discrete'>Discrete</a></li> <li><a href='#topological'>Topological</a></li> </ul> <li><a href='#terminology'>Terminology</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#on_hilbert_spaces'>On Hilbert spaces</a></li> </ul> </ul> </div> <h2 id="definition">Definition</h2> <h3 id="discrete">Discrete</h3> <p>A <strong><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>-linear map</strong> (also <strong><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>-linear function</strong>, <strong><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>-<a class="existingWikiWord" href="/nlab/show/linear+operator">linear operator</a></strong>, or <strong><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>-linear transformation</strong>) is a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> in <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>-<a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> (or <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>-<a class="existingWikiWord" href="/nlab/show/Mod">Mod</a>), that is a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> (or <a class="existingWikiWord" href="/nlab/show/modules">modules</a>). Often one suppresses mention of the <a class="existingWikiWord" href="/nlab/show/field">field</a> (or <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> or <a class="existingWikiWord" href="/nlab/show/rig">rig</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">K</annotation></semantics></math>.</p> <p>In elementary terms, a <strong><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>-linear map</strong> between <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>-<a class="existingWikiWord" href="/nlab/show/linear+spaces">linear spaces</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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo lspace="verythinmathspace">:</mo><mi>V</mi><mo>→</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">T\colon V \to W</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>r</mi><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>r</mi><mi>T</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>T</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex"> T(r x + y) = r T(x) + T(y) ,</annotation></semantics></math></div> <p>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> 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> elements 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> 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">r</annotation></semantics></math> an element of <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>. (It is an easy exercise that this one identity is enough to ensure that <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> preserves all <a class="existingWikiWord" href="/nlab/show/linear+combination"> linear combinations</a>.)</p> <h3 id="topological">Topological</h3> <p>The <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> between <a class="existingWikiWord" href="/nlab/show/topological+vector+spaces">topological vector spaces</a> are of course the <a class="existingWikiWord" href="/nlab/show/continuous+map">continuous</a> linear maps. Between <a class="existingWikiWord" href="/nlab/show/Banach+spaces">Banach spaces</a> (including of course <a class="existingWikiWord" href="/nlab/show/Hilbert+spaces">Hilbert spaces</a>), these are the same as the <a class="existingWikiWord" href="/nlab/show/bounded+map">bounded</a> linear maps, so they're often called <strong><a class="existingWikiWord" href="/nlab/show/bounded+operators">bounded operators</a></strong> (with linearity tacitly assumed).</p> <p>In this context, <strong><a class="existingWikiWord" href="/nlab/show/linear+operators">linear operators</a></strong> are more general; they are (in general) only <a class="existingWikiWord" href="/nlab/show/partial+functions">partial functions</a>. However, we still require the <a class="existingWikiWord" href="/nlab/show/domain">domain</a> of the partial function to be a linear <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a>, after which the definition above applies. Because one typically restricts attention to <a class="existingWikiWord" href="/nlab/show/complete+spaces">complete spaces</a>, the <span class="newWikiWord"> densely-defined operators<a href="/nlab/new/densely-defined+operator">?</a></span> (where the domain is a <a class="existingWikiWord" href="/nlab/show/dense+subspace">dense subspace</a>) are the most general needed. To specify that the domain of a linear operator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo lspace="verythinmathspace">:</mo><mi>V</mi><mo>→</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">T\colon V \to W</annotation></semantics></math> is all 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>, one may use a non-‘operator’ term, such as <strong>linear mapping</strong>.</p> <p>Notice that we do <em>not</em> require partially-defined linear operators to be continuous; see <a class="existingWikiWord" href="/nlab/show/unbounded+operator">unbounded operator</a>. However, we have the theorem that any densely-defined, continuous, linear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo lspace="verythinmathspace">:</mo><mi>V</mi><mo>→</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">T\colon V \to W</annotation></semantics></math>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/complete+space">complete</a> and <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff</a>, extends uniquely to all 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>. Thus one typically assumes that a continuous (or bounded) linear operator <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> is defined on all 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> while an arbitrary linear operator <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> is defined only on a dense subspace 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>.</p> <h2 id="terminology">Terminology</h2> <p>In elementary mathematics (at least as taught in the United States, perhaps elsewhere?), the term ‘linear function’ is usually used more generally, for an <a class="existingWikiWord" href="/nlab/show/affine+map">affine map</a> (but still between vector spaces); in this same context, the term ‘linear transformation’ is often used instead for specifically linear maps. (Another difference at this level is that ‘linear functions’ are usually scalar-valued, while ‘linear transformations’ are usually vector-valued.)</p> <p>In <span class="newWikiWord">operator theory<a href="/nlab/new/operator+theory">?</a></span>, one sometimes distinguishes ‘linear maps’ (defined everywhere, but not necessarily continuous in general) from ‘linear operators’ (partially defined in general, but assumed to be defined everywhere if continuous between complete Hausdorff spaces). There is also a tendency for ‘operator’ to be used only for (possibly partial) <a class="existingWikiWord" href="/nlab/show/endomorphisms">endomorphisms</a>, that is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo lspace="verythinmathspace">:</mo><mi>V</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">T\colon V \to V</annotation></semantics></math>; then operators may be <a class="existingWikiWord" href="/nlab/show/composition">composed</a>, giving rise to an <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/function+space">function space</a>, then an endomorphism 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> is an <a class="existingWikiWord" href="/nlab/show/operator">operator</a> in the sense of <a class="existingWikiWord" href="/nlab/show/higher-order+logic">higher-order logic</a>; the more general meaning of ‘linear operator’ is abstracted from this.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/function">function</a>, <a class="existingWikiWord" href="/nlab/show/quadratic+function">quadratic function</a>, <a class="existingWikiWord" href="/nlab/show/polynomial+function">polynomial function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+product">operator product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+isomorphism">linear isomorphism</a>, <a class="existingWikiWord" href="/nlab/show/linear+isometry">linear isometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bounded+linear+operator">bounded linear operator</a> / <a class="existingWikiWord" href="/nlab/show/unbounded+linear+operator">unbounded linear operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/eigenvector">eigenvector</a>, <a class="existingWikiWord" href="/nlab/show/eigenvalue">eigenvalue</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/antilinear+map">antilinear map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/self-adjoint+operator">self-adjoint operator</a>, <a class="existingWikiWord" href="/nlab/show/positive+operator">positive operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+subspace">linear subspace</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+differential+equation">linear differential equation</a></p> </li> <li> <p><span class="newWikiWord">piecewise linear function<a href="/nlab/new/piecewise+linear+function">?</a></span></p> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>Textbook accounts:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Igor+R.+Shafarevich">Igor R. Shafarevich</a>, <a class="existingWikiWord" href="/nlab/show/Alexey+O.+Remizov">Alexey O. Remizov</a>: §4 in: <em>Linear Algebra and Geometry</em> (2012) [<a href="https://doi.org/10.1007/978-3-642-30994-6">doi:10.1007/978-3-642-30994-6</a>, <a href="https://maa.org/press/maa-reviews/linear-algebra-and-geometry">MAA-review</a>]</li> </ul> <h3 id="on_hilbert_spaces">On Hilbert spaces</h3> <p>On linear operators on <a class="existingWikiWord" href="/nlab/show/Hilbert+spaces">Hilbert spaces</a> (see also at <em><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a></em>):</p> <p>Lecture notes:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Bergfinnur+Durhuus">Bergfinnur Durhuus</a>, <em>Operators on Hilbert Space</em> [<a href="https://web.math.ku.dk/~durhuus/MatFys/MatFys4.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Durhuus-OperatorsOnHilbertSpace.pdf" title="pdf">pdf</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 12, 2023 at 22:04:13. 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