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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Nullfunktion</span></h1>

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<p>Die <b>Nullfunktion</b> ist in der <a href="Mathematik" title="Mathematik">Mathematik</a>, insbesondere der <a href="Analysis" title="Analysis">Analysis</a>, eine <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktion</a>, deren <a href="Funktionswert" class="mw-redirect" title="Funktionswert">Funktionswert</a> unabhängig vom übergebenen Wert immer die Zahl <a href="Null" title="Null">Null</a> ist. Allgemeiner ist die <b>Nullabbildung</b> oder der <b>Nulloperator</b> in der <a href="Lineare_Algebra" title="Lineare Algebra">linearen Algebra</a> eine Abbildung zwischen zwei <a href="Vektorraum" title="Vektorraum">Vektorräumen</a>, die stets den <a href="Nullvektor" title="Nullvektor">Nullvektor</a> des <a href="Zielmenge" title="Zielmenge">Zielraums</a> ergibt. Noch allgemeiner wird die Nullabbildung in der <a href="Algebra" title="Algebra">Algebra</a> gefasst und dort ist sie eine Abbildung von einer beliebigen <a href="Menge_(Mathematik)" title="Menge (Mathematik)">Menge</a> in eine Menge, auf der eine <a href="Verkn%C3%BCpfung_(Mathematik)" title="Verknüpfung (Mathematik)">Verknüpfung</a> mit <a href="Neutrales_Element" title="Neutrales Element">neutralem Element</a> definiert ist, die immer dieses neutrale Element ergibt. Die Nullfunktion hat viele Eigenschaften und wird in der Mathematik oft als Beispiel oder als Gegenbeispiel verwendet. Sie ist die <a href="Trivial" class="mw-redirect" title="Trivial">triviale</a> <a href="L%C3%B6sung_(Mathematik)" title="Lösung (Mathematik)">Lösung</a> einer Reihe mathematischer <a href="Problem" title="Problem">Probleme</a>, wie zum Beispiel <a href="Lineare_Gleichung#Homogenität" title="Lineare Gleichung">homogener</a> <a href="Lineare_Differentialgleichung" class="mw-redirect" title="Lineare Differentialgleichung">linearer Differentialgleichungen</a> und <a href="Integralgleichung" title="Integralgleichung">Integralgleichungen</a>.

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<div class="mw-heading mw-heading2"><h2 id="Reelle_Nullfunktion">Reelle Nullfunktion</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Definition">Definition</h3></div>

<p>In der reellen <a href="Analysis" title="Analysis">Analysis</a> ist die Nullfunktion die reelle <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktion</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \colon \mathbb {R} \to \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \phi \colon \mathbb {R} \to \mathbb {R} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21d9e816cc7ebd2000448cc674f2b9bfef6a938c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.39ex; height:2.509ex;" alt="{\displaystyle \phi \colon \mathbb {R} \to \mathbb {R} }" loading="lazy"></span>, die jedem Argument die Zahl <a href="Null" title="Null">Null</a> zuordnet, das heißt, es gilt

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<annotation encoding="application/x-tex">{\displaystyle \phi (x)=0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ed9b3798a5d330ba68f16e02e5b4063d036070ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.785ex; height:2.843ex;" alt="{\displaystyle \phi (x)=0}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9c6d458566aec47a7259762034790c8981aefab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle x\in \mathbb {R} }" loading="lazy"></span>. Mit Hilfe des <a href="Gleichheitszeichen" title="Gleichheitszeichen">Identitätssymbols</a> wird die Nullfunktion auch durch

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<annotation encoding="application/x-tex">{\displaystyle \phi \equiv 0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6e007b48610ca218bdb9c000cb963208bdd9eb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.646ex; height:2.509ex;" alt="{\displaystyle \phi \equiv 0}" loading="lazy"></span></dd></dl>

<p>notiert. Der <a href="Funktionsgraph" title="Funktionsgraph">Graph</a> der Nullfunktion ist die gesamte <a href="X-Achse" class="mw-redirect" title="X-Achse">x-Achse</a>. Gelegentlich wird der <a href="Definitionsmenge" title="Definitionsmenge">Definitionsbereich</a> der Nullfunktion auch auf eine <a href="Teilmenge" title="Teilmenge">Teilmenge</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega \subset \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \Omega \subset \mathbb {R} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e66ab6d255b093f9b592bd1887a745dc304d4d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.455ex; height:2.176ex;" alt="{\displaystyle \Omega \subset \mathbb {R} }" loading="lazy"></span> eingeschränkt.

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<div class="mw-heading mw-heading3"><h3 id="Eigenschaften">Eigenschaften</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Einordnung">Einordnung</h4></div>

<p>Die Nullfunktion ist ein Spezialfall folgender Funktionenklassen:

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<ul><li>Sie ist eine spezielle <a href="Konstante_Funktion" title="Konstante Funktion">konstante Funktion</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=c}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)=c}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/890f66f7bf07cf23de8c877eacf5c65a52e01f4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.523ex; height:2.843ex;" alt="{\displaystyle f(x)=c}" loading="lazy"></span>, und zwar gerade diejenige, deren Konstante <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=0}">

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<annotation encoding="application/x-tex">{\displaystyle c=0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9ee918699d0cb4b8c633cc1f520a8a7a174f44a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c=0}" loading="lazy"></span> ist.</li>

<li>Sie ist eine spezielle <a href="Lineare_Funktion" title="Lineare Funktion">lineare Funktion</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=mx+b}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)=mx+b}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/554c3e2a5be5528277a5668ec3d0b15adeaa0ad1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.724ex; height:2.843ex;" alt="{\displaystyle f(x)=mx+b}" loading="lazy"></span>, und zwar diejenige, deren <a href="Steigung" title="Steigung">Steigung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=0}">

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<annotation encoding="application/x-tex">{\displaystyle m=0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e57f21007575fd03e3be0da20af34d25829cc9a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=0}" loading="lazy"></span> und <a href="Ordinatenabschnitt" class="mw-redirect" title="Ordinatenabschnitt">Ordinatenabschnitt</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=0}">

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<annotation encoding="application/x-tex">{\displaystyle b=0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19206e7d4dab695ccb34c502eff0741e98dbdfc2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b=0}" loading="lazy"></span> sind.</li>

<li>Sie ist eine spezielle <a href="Polynom" title="Polynom">Polynomfunktion</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{1}x+a_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{1}x+a_{0}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a8df67ee0fb103d19e0109c17c77bba1a7a15af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.693ex; height:3.176ex;" alt="{\displaystyle f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{1}x+a_{0}}" loading="lazy"></span>, nämlich das <a href="Nullpolynom" class="mw-redirect" title="Nullpolynom">Nullpolynom</a>, bei dem alle Koeffizienten <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}=0}">

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<annotation encoding="application/x-tex">{\displaystyle a_{i}=0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1d2e283d81c74e9c6742dcbbcad8c622ef0c3c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.29ex; height:2.509ex;" alt="{\displaystyle a_{i}=0}" loading="lazy"></span> sind. Der <a href="Grad_(Polynom)" title="Grad (Polynom)">Grad</a> des Nullpolynoms wird meist nicht als <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

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<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"></span>, sondern als <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }">

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<annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }" loading="lazy"></span> definiert.</li></ul>

<div class="mw-heading mw-heading4"><h4 id="Symmetrien">Symmetrien</h4></div>

<p>Die Nullfunktion ist als einzige Funktion gleichzeitig <a href="Gerade_und_ungerade_Funktionen" title="Gerade und ungerade Funktionen">gerade und ungerade</a>, das heißt, es gilt

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<annotation encoding="application/x-tex">{\displaystyle \phi (x)=\phi (-x)=-\phi (x)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/25e762b1f8f12dafeec015c08f47aaf0a33e89e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.386ex; height:2.843ex;" alt="{\displaystyle \phi (x)=\phi (-x)=-\phi (x)}" loading="lazy"></span>.</dd></dl>

<p>Weiter ist sie weder <a href="Positive_und_negative_Zahlen" title="Positive und negative Zahlen">positiv noch negativ</a>, stattdessen ist sie sowohl nichtpositiv als auch nichtnegativ, also

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<annotation encoding="application/x-tex">{\displaystyle \phi (x)\leq 0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/164ac981c883425f4a31d81c328ac2f7f4727989.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.785ex; height:2.843ex;" alt="{\displaystyle \phi (x)\leq 0}" loading="lazy"></span>   und   <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (x)\geq 0}">

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<annotation encoding="application/x-tex">{\displaystyle \phi (x)\geq 0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f05945d97876f27c02e6c40ad59849f7467ae9ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.785ex; height:2.843ex;" alt="{\displaystyle \phi (x)\geq 0}" loading="lazy"></span>.</dd></dl>

<p>Die <a href="Nullstelle" title="Nullstelle">Nullstellen</a> der Nullfunktion sind damit alle Zahlen der Definitionsmenge und ihre <a href="Nichtnullstellenmenge" title="Nichtnullstellenmenge">Nichtnullstellenmenge</a> ist demnach <a href="Leere_Menge" title="Leere Menge">leer</a>. Das Minimum und das Maximum der Nullfunktion sind ebenfalls Null:

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<annotation encoding="application/x-tex">{\displaystyle \max _{x\in \mathbb {R} }\phi (x)=\min _{x\in \mathbb {R} }\phi (x)=0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8fd5f8878a78678438a76fa39f4a9d335d93c0d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:25.383ex; height:4.176ex;" alt="{\displaystyle \max _{x\in \mathbb {R} }\phi (x)=\min _{x\in \mathbb {R} }\phi (x)=0}" loading="lazy"></span>.</dd></dl>

<p>Weiterhin ist die Nullfunktion, wie jede konstante Funktion, gleichzeitig <a href="Reelle_monotone_Funktion" class="mw-redirect" title="Reelle monotone Funktion">monoton steigend und fallend</a> (jedoch nicht streng) und, wie jede lineare Funktion, gleichzeitig <a href="Konvexe_und_konkave_Funktionen" title="Konvexe und konkave Funktionen">konvex und konkav</a>.

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<div class="mw-heading mw-heading4"><h4 id="Ableitungen">Ableitungen</h4></div>

<p>Die Nullfunktion ist eine <a href="Glatte_Funktion" title="Glatte Funktion">glatte Funktion</a>, also beliebig oft <a href="Stetige_Funktion" title="Stetige Funktion">stetig</a> <a href="Differenzierbarkeit" title="Differenzierbarkeit">differenzierbar</a>, wobei jede ihrer <a href="Differentialrechnung" title="Differentialrechnung">Ableitungen</a> wieder die Nullfunktion selbst ist, das heißt

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<annotation encoding="application/x-tex">{\displaystyle \phi ^{(n)}(x)=\phi (x)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d84b1a34ab538fc359c6684415f88528a8c539c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.645ex; height:3.343ex;" alt="{\displaystyle \phi ^{(n)}(x)=\phi (x)}" loading="lazy"></span></dd></dl>

<p>für jedes <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }" loading="lazy"></span>. Neben den Vielfachen der <a href="Exponentialfunktion" title="Exponentialfunktion">Exponentialfunktion</a> ist die Nullfunktion die einzige Funktion mit dieser Eigenschaft. Die Nullfunktion selbst ist wiederum die Ableitung einer konstanten Funktion und allgemein die <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n+1)}">

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<annotation encoding="application/x-tex">{\displaystyle (n+1)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b30a29cfd35628469f9dbffea4804f5b422f3037.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="{\displaystyle (n+1)}" loading="lazy"></span>-te Ableitung eines Polynoms vom Grad <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>.

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<div class="mw-heading mw-heading4"><h4 id="Integral">Integral</h4></div>

<p>Das <a href="Integralrechnung" title="Integralrechnung">Integral</a> der Nullfunktion ergibt unabhängig von den Integrationsgrenzen immer Null, also

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<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}\phi (x)~dx=0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7df8cc4055c33c5c0f66bedc4d2f72e852213a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.7ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}\phi (x)~dx=0}" loading="lazy"></span>.</dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle a,b\in \mathbb {R} \cup \{-\infty ,\infty \}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d7dbe90d106d3d9bde60b74e0c6778a8e9dd19c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.177ex; height:2.843ex;" alt="{\displaystyle a,b\in \mathbb {R} \cup \{-\infty ,\infty \}}" loading="lazy"></span>. Die Nullfunktion ist damit die einzige Polynomfunktion, die über den gesamten reellen Zahlen integrierbar ist. <a href="Stammfunktion" title="Stammfunktion">Stammfunktion</a> der Nullfunktion ist die Nullfunktion selbst und, da die <a href="Integrationskonstante" class="mw-redirect" title="Integrationskonstante">Integrationskonstante</a> frei wählbar ist, auch jede konstante Funktion.

</p>

<div class="mw-heading mw-heading4"><h4 id="Lösung_von_Gleichungen"><span id="L.C3.B6sung_von_Gleichungen"></span>Lösung von Gleichungen</h4></div>

<p>Die Nullfunktion ist die triviale Lösung der vier <a href="Funktionalgleichung#Von_Cauchy_untersuchte_Funktionalgleichungen" title="Funktionalgleichung">Cauchy-Funktionalgleichungen</a>:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f(x+y)&=f(x)+f(y)\\f(x+y)&=f(x)\cdot f(y)\\f(x\cdot y)&=f(x)+f(y)\\f(x\cdot y)&=f(x)\cdot f(y)\\\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/911c5dd95aba707728272132022a90a27569a646.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:23.765ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}f(x+y)&=f(x)+f(y)\\f(x+y)&=f(x)\cdot f(y)\\f(x\cdot y)&=f(x)+f(y)\\f(x\cdot y)&=f(x)\cdot f(y)\\\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>Weiter löst die Nullfunktion jede homogene <a href="Lineare_gew%C3%B6hnliche_Differentialgleichung" title="Lineare gewöhnliche Differentialgleichung">lineare Differentialgleichung</a> der Form

</p>

<msub>

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<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle a_{n}(x)f^{(n)}(x)+a_{n-1}(x)f^{(n-1)}+\dotsb +a_{1}(x)f'(x)+a_{0}(x)f(x)=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/651406b8c66362c9efd404f4f8b1d4807ed69d07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:64.905ex; height:3.343ex;" alt="{\displaystyle a_{n}(x)f^{(n)}(x)+a_{n-1}(x)f^{(n-1)}+\dotsb +a_{1}(x)f'(x)+a_{0}(x)f(x)=0}" loading="lazy"></span></dd></dl>

<p>und jede homogene lineare <a href="Integralgleichung" title="Integralgleichung">Integralgleichung</a> der Art

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<annotation encoding="application/x-tex">{\displaystyle \lambda f(x)+\int _{a}^{x}K(x,y)f(y)~dy=0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c53070eee0f89a68964b3e8f6a6186ac1f9337c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.488ex; height:5.843ex;" alt="{\displaystyle \lambda f(x)+\int _{a}^{x}K(x,y)f(y)~dy=0}" loading="lazy"></span></dd></dl>

<p>mit <a href="Integralkern" class="mw-redirect" title="Integralkern">Integralkern</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(x,y)}">

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<annotation encoding="application/x-tex">{\displaystyle K(x,y)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b4554c1b79ecaae2584d99cc3959f2eed7e6d41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.394ex; height:2.843ex;" alt="{\displaystyle K(x,y)}" loading="lazy"></span> und Vorfaktor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

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<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span>. Umgekehrt wird eine inhomogene lineare Differential- oder Integralgleichung nie durch die Nullfunktion gelöst.

</p>

<div class="mw-heading mw-heading2"><h2 id="Nullabbildungen_zwischen_Vektorräumen"><span id="Nullabbildungen_zwischen_Vektorr.C3.A4umen"></span>Nullabbildungen zwischen Vektorräumen</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Definition_2">Definition</h3></div>

<p>In der <a href="Lineare_Algebra" title="Lineare Algebra">linearen Algebra</a> heißt eine Abbildung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \colon V\to W}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \phi \colon V\to W}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc558648a9e3bc4824daafd085c9526e942868ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.256ex; height:2.509ex;" alt="{\displaystyle \phi \colon V\to W}" loading="lazy"></span> zwischen zwei <a href="Vektorraum" title="Vektorraum">Vektorräumen</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> über dem gleichen <a href="K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körper</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span> Nullabbildung oder Nulloperator, wenn für alle Vektoren <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v\in V}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99886ebbde63daa0224fb9bf56fa11b3c8a6f4fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.756ex; height:2.176ex;" alt="{\displaystyle v\in V}" loading="lazy"></span>

</p>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \phi (v)=0_{W}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/340a931d5ddb9e1ba4bb18fb0de0ccc8c38d200d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.537ex; height:2.843ex;" alt="{\displaystyle \phi (v)=0_{W}}" loading="lazy"></span></dd></dl>

<p>gilt, wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0_{W}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0_{W}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5db3f42738ae643d3d698dcdd7a41900303963e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.117ex; height:2.509ex;" alt="{\displaystyle 0_{W}}" loading="lazy"></span> der eindeutig bestimmte <a href="Nullvektor" title="Nullvektor">Nullvektor</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> ist. Gelegentlich wird die Nullabbildung auch direkt durch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"></span> notiert, sofern aus dem Kontext klar ist, ob die Nullabbildung oder die Zahl Null gemeint ist. Auch hier kann der Definitionsbereich der Nullabbildung auf eine Teilmenge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subset V}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle U\subset V}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52e93662b421bc57d0c605ca5d31007b234e68b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.668ex; height:2.176ex;" alt="{\displaystyle U\subset V}" loading="lazy"></span> eingeschränkt werden.

</p>

<div class="mw-heading mw-heading3"><h3 id="Beispiele">Beispiele</h3></div>

<ul><li>die reelle Nullfunktion des vorangegangenen Abschnitts und allgemeiner reelle oder <a href="Komplexe_Zahl" title="Komplexe Zahl">komplexe</a> Funktionen ein oder mehrerer Variablen, deren Funktionswert die Zahl Null oder der Nullvektor ist</li>

<li>jede Abbildung von einem beliebigen Vektorraum <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> in den <a href="Nullvektorraum" title="Nullvektorraum">Nullvektorraum</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{0\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ff0df9ef65c0572eb676580ce1c02b8ec40f694.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{0\}}" loading="lazy"></span> und jede <a href="Lineare_Abbildung" title="Lineare Abbildung">lineare Abbildung</a> vom Nullvektorraum in einen beliebigen Vektorraum <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li>

<li>eine quadratische Matrix, die in ihr <a href="Charakteristisches_Polynom" title="Charakteristisches Polynom">charakteristisches Polynom</a> eingesetzt wird, nach dem <a href="Satz_von_Cayley-Hamilton" title="Satz von Cayley-Hamilton">Satz von Cayley-Hamilton</a><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li>

<li>die <a href="Determinantenfunktion" title="Determinantenfunktion">Determinantenfunktion</a> auf der Menge der <a href="Singul%C3%A4re_Matrix" class="mw-redirect" title="Singuläre Matrix">singulären</a> quadratischen <a href="Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrizen</a><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Eigenschaften_2">Eigenschaften</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Linearität"><span id="Linearit.C3.A4t"></span>Linearität</h4></div>

<p>Die Nullabbildung ist eine <a href="Lineare_Abbildung" title="Lineare Abbildung">lineare Abbildung</a>, also ein Vektorraumhomomorphismus, das heißt, es gilt

</p>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \phi (av+bw)=a\phi (v)+b\phi (w)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19e11127b771f4c38ceeda37848d83bda1b40443.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.402ex; height:2.843ex;" alt="{\displaystyle \phi (av+bw)=a\phi (v)+b\phi (w)}" loading="lazy"></span></dd></dl>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle v,w\in V}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4799d1b026506616abad8de636f46738b8a55a96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.454ex; height:2.509ex;" alt="{\displaystyle v,w\in V}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a,b\in K}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3ff3a49d65fc590e33a74fd613900dd5924d6ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.168ex; height:2.509ex;" alt="{\displaystyle a,b\in K}" loading="lazy"></span>. Sie liegt also im <a href="Lineare_Abbildung#Vektorraum_der_linearen_Abbildungen" title="Lineare Abbildung">Vektorraum der linearen Abbildungen</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(V,W)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(V,W)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6e7784b18512cb31d0952dab18cc9612caea99e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.649ex; height:2.843ex;" alt="{\displaystyle L(V,W)}" loading="lazy"></span> und ist dort selbst der Nullvektor.

</p><p>Jede Nullabbildung zwischen endlichdimensionalen Vektorräumen wird bezüglich beliebiger Basen durch eine <a href="Nullmatrix" title="Nullmatrix">Nullmatrix</a> der Größe <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim W\times \dim V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \dim W\times \dim V}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ffb404c09ca564f69531e1d010d63c8db575da7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.588ex; height:2.176ex;" alt="{\displaystyle \dim W\times \dim V}" loading="lazy"></span> dargestellt.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Ihr <a href="Kern_(Algebra)" title="Kern (Algebra)">Kern</a> ist ganz <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span>, ihr <a href="Bild_(Mathematik)" title="Bild (Mathematik)">Bild</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0_{W}\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{0_{W}\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f1f417fc8c67fa850ba731083a52d6e472319c22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.442ex; height:2.843ex;" alt="{\displaystyle \{0_{W}\}}" loading="lazy"></span> und somit ihr <a href="Rang_(Lineare_Algebra)" title="Rang (Lineare Algebra)">Rang</a> immer <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"></span>. Ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V=W}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/740038d36bd79466d6938d73b83fe737161fa1c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.321ex; height:2.176ex;" alt="{\displaystyle V=W}" loading="lazy"></span>, dann ist besitzt die Nullabbildung als einzigen <a href="Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwert</a> die Zahl Null und der zugehörige Eigenraum ist ganz <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading4"><h4 id="Operatornorm">Operatornorm</h4></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> <a href="Normierter_Raum" title="Normierter Raum">normierte Räume</a> mit jeweiligen <a href="Norm_(Mathematik)" title="Norm (Mathematik)">Normen</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|_{V}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{V}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a94815a55b53ec1446f1f7df2afc4fdd55038ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.5ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|_{V}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|_{W}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{W}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1ac0304adb0b7e9ba2c8aae7233b46a55947294.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.958ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|_{W}}" loading="lazy"></span>, dann ist die <a href="Operatornorm" title="Operatornorm">Operatornorm</a> der Nullabbildung

</p>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|\phi \|=\sup _{\|v\|_{V}=1}\|\phi (v)\|_{W}=\|0_{W}\|_{W}=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be41910357350560eac96a9ad3a26235c20defa9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.285ex; height:5.009ex;" alt="{\displaystyle \|\phi \|=\sup _{\|v\|_{V}=1}\|\phi (v)\|_{W}=\|0_{W}\|_{W}=0}" loading="lazy"></span>.</dd></dl>

<p>Die Nullabbildung selbst stellt für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W=\mathbb {R} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd5faedf8fa728005dbc8a5f662e7b286a80ae97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.212ex; height:2.176ex;" alt="{\displaystyle W=\mathbb {R} }" loading="lazy"></span> eine <a href="Halbnorm" title="Halbnorm">Halbnorm</a> dar.

</p>

<div class="mw-heading mw-heading4"><h4 id="Lösung_von_Gleichungen_2"><span id="L.C3.B6sung_von_Gleichungen_2"></span>Lösung von Gleichungen</h4></div>

<p>Allgemein löst die Nullabbildung jede homogene lineare Operatorgleichung

</p>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}u=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d09ca579e5170e41f808274e2325c3344d93748.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.194ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}u=0}" loading="lazy"></span>,</dd></dl>

<p>wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}\in L(V,W)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}\in L(V,W)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f1376f521a906487422100cce113bfdafdfd1e8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.093ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}\in L(V,W)}" loading="lazy"></span> ein <a href="Linearer_Operator" title="Linearer Operator">linearer Operator</a> ist, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"></span> die gesuchte Funktion und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"></span> die Nullfunktion ist. Umgekehrt wird eine inhomogene lineare Operatorgleichung, bei der also die rechte Seite ungleich der Nullfunktion ist, nie durch die Nullabbildung gelöst.

</p>

<div class="mw-heading mw-heading2"><h2 id="Nullabbildungen_in_ein_Magma_mit_Eins">Nullabbildungen in ein Magma mit Eins</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Definition_3">Definition</h3></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> eine <a href="Menge_(Mathematik)" title="Menge (Mathematik)">Menge</a> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> ein <a href="Magma_(Mathematik)" title="Magma (Mathematik)">Magma</a> mit Eins, das heißt eine Menge versehen mit einer zweistelligen <a href="Verkn%C3%BCpfung_(Mathematik)" title="Verknüpfung (Mathematik)">Verknüpfung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ast }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ast }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f1858484bef51b1435c2b986c728a81788051803.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \ast }" loading="lazy"></span> mit <a href="Neutrales_Element" title="Neutrales Element">neutralem Element</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"></span>, dann heißt eine Abbildung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \colon X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi \colon X\to Y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/980fd1d353c89457722d77bfa1d61d63443990fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.787ex; height:2.509ex;" alt="{\displaystyle \phi \colon X\to Y}" loading="lazy"></span> Nullabbildung, wenn für alle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"></span>

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi (x)=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ed9b3798a5d330ba68f16e02e5b4063d036070ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.785ex; height:2.843ex;" alt="{\displaystyle \phi (x)=0}" loading="lazy"></span></dd></dl>

<p>gilt. Wichtige Beispiele für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Y,\ast )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (Y,\ast )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/329d7744671f11dc954a0a8b5a52428bf42b1bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.779ex; height:2.843ex;" alt="{\displaystyle (Y,\ast )}" loading="lazy"></span> sind <a href="Monoid" title="Monoid">Monoide</a>, <a href="Gruppe_(Mathematik)" title="Gruppe (Mathematik)">Gruppen</a>, <a href="Ring_(Algebra)" title="Ring (Algebra)">Ringe</a>, <a href="Modul_(Mathematik)" title="Modul (Mathematik)">Moduln</a> und – wie im vorangegangenen Abschnitt – Vektorräume.

</p>

<div class="mw-heading mw-heading3"><h3 id="Beispiele_2">Beispiele</h3></div>

<ul><li>die <a href="Boolesche_Funktion" title="Boolesche Funktion">boolesche Funktion</a> der <a href="Kontradiktion" title="Kontradiktion">Kontradiktion</a> in einen <a href="Boolescher_Ring" class="mw-redirect" title="Boolescher Ring">booleschen Ring</a> bzw. eine <a href="Boolesche_Algebra" title="Boolesche Algebra">boolesche Algebra</a></li>

<li>die <a href="Polynomfunktion" class="mw-redirect" title="Polynomfunktion">Polynomfunktion</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{q}-x}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{q}-x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d92f4fb8a44488fa3c3570a2276311f9d0e81b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.488ex; height:2.509ex;" alt="{\displaystyle x^{q}-x}" loading="lazy"></span> in einem <a href="Polynomring" title="Polynomring">Polynomring</a> über einem <a href="Endlicher_K%C3%B6rper" title="Endlicher Körper">endlichen Körper</a> mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"></span> Elementen<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>-te Potenz einer <a href="Nilpotenz" class="mw-redirect" title="Nilpotenz">nilpotenten</a> Abbildung in einen Ring, wenn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> größer oder gleich dem <a href="Nilpotenzindex" class="mw-redirect" title="Nilpotenzindex">Nilpotenzindex</a> der Abbildung ist<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li>

<li>das <a href="Ma%C3%9F_(Mathematik)" title="Maß (Mathematik)">Nullmaß</a>, das jeder Menge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> den Wert <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (A)=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (A)=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eeb31cb69e717997b331a32910b5bb51b475df30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.215ex; height:2.843ex;" alt="{\displaystyle \mu (A)=0}" loading="lazy"></span> zuordnet</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Eigenschaften_3">Eigenschaften</h3></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> zwei Magmen, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> mit Eins, dann ist die Nullabbildung ein <a href="Magmenhomomorphismus" class="mw-redirect" title="Magmenhomomorphismus">Magmenhomomorphismus</a>.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> zwei Ringe, dann ist die Nullabbildung ein <a href="Ringhomomorphismus" title="Ringhomomorphismus">Ringhomomorphismus</a>. Ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> ein <a href="Einfacher_Ring" class="mw-redirect" title="Einfacher Ring">einfacher Ring</a> (beispielsweise ein Körper oder ein <a href="Schiefk%C3%B6rper" title="Schiefkörper">Schiefkörper</a>), dann ist jeder Ringhomomorphismus entweder <a href="Injektivit%C3%A4t" class="mw-redirect" title="Injektivität">injektiv</a> oder die Nullabbildung.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> zwei Moduln, dann ist die Nullabbildung ein <a href="Modulhomomorphismus" title="Modulhomomorphismus">Modulhomomorphismus</a>.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> zwei <a href="Algebra_%C3%BCber_einem_kommutativen_Ring" title="Algebra über einem kommutativen Ring">Algebren über einem Ring</a>, dann ist die Nullabbildung ein <a href="Algebrenhomomorphismus" class="mw-redirect" title="Algebrenhomomorphismus">Algebrenhomomorphismus</a>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Annihilator_(Mathematik)" title="Annihilator (Mathematik)">Annihilator (Mathematik)</a></li>

<li><a href="Nullring" title="Nullring">Nullring</a></li>

<li><a href="Nullteiler" title="Nullteiler">Nullteiler</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>Martin Barner, Friedrich Flohr: <cite style="font-style:italic">Analysis I</cite>. de Gruyter, 2000, ISBN 3-11-016778-6.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Nullfunktion&rft.au=Martin+Barner%2C+Friedrich+Flohr&rft.btitle=Analysis+I&rft.date=2000&rft.genre=book&rft.isbn=3110167786&rft.pub=de+Gruyter" style="display:none"> </span></li>

<li><a href="Siegfried_Bosch" title="Siegfried Bosch">Siegfried Bosch</a>: <cite style="font-style:italic">Lineare Algebra</cite>. Springer, 2009, ISBN 3-540-76437-2.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Nullfunktion&rft.au=Siegfried+Bosch&rft.btitle=Lineare+Algebra&rft.date=2009&rft.genre=book&rft.isbn=3540764372&rft.pub=Springer" style="display:none"> </span></li>

<li>Christian Karpfinger, Kurt Meyberg: <cite style="font-style:italic">Algebra: Gruppen – Ringe – Körper</cite>. Springer, 2008, ISBN 3-8274-2018-0.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Nullfunktion&rft.au=Christian+Karpfinger%2C+Kurt+Meyberg&rft.btitle=Algebra%3A+Gruppen+-+Ringe+-+K%C3%B6rper&rft.date=2008&rft.genre=book&rft.isbn=3827420180&rft.pub=Springer" style="display:none"> </span></li>

<li><a href="Gilbert_Strang" title="Gilbert Strang">Gilbert Strang</a>: <cite style="font-style:italic">Lineare Algebra</cite>. Springer, 2003, ISBN 3-540-43949-8.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Nullfunktion&rft.au=Gilbert+Strang&rft.btitle=Lineare+Algebra&rft.date=2003&rft.genre=book&rft.isbn=3540439498&rft.pub=Springer" style="display:none"> </span></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Barner, Flohr: <cite style="font-style:italic">Analysis I</cite>. <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>247</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Nullfunktion&rft.au=Barner%2C+Flohr&rft.btitle=Analysis+I&rft.genre=book&rft.pages=247" style="display:none"> </span></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Bosch: <cite style="font-style:italic">Lineare Algebra</cite>. <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>78</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Nullfunktion&rft.au=Bosch&rft.btitle=Lineare+Algebra&rft.genre=book&rft.pages=78" style="display:none"> </span></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Bosch: <cite style="font-style:italic">Lineare Algebra</cite>. <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>204</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Nullfunktion&rft.au=Bosch&rft.btitle=Lineare+Algebra&rft.genre=book&rft.pages=204" style="display:none"> </span></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">Bosch: <cite style="font-style:italic">Lineare Algebra</cite>. <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>141</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Nullfunktion&rft.au=Bosch&rft.btitle=Lineare+Algebra&rft.genre=book&rft.pages=141" style="display:none"> </span></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Bosch: <cite style="font-style:italic">Lineare Algebra</cite>. <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>93</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Nullfunktion&rft.au=Bosch&rft.btitle=Lineare+Algebra&rft.genre=book&rft.pages=93" style="display:none"> </span></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text">Karpfinger, Meyberg: <cite style="font-style:italic">Algebra: Gruppen – Ringe – Körper</cite>. <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>158</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Nullfunktion&rft.au=Karpfinger%2C+Meyberg&rft.btitle=Algebra%3A+Gruppen+-+Ringe+-+K%C3%B6rper&rft.genre=book&rft.pages=158" style="display:none"> </span></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text">Karpfinger, Meyberg: <cite style="font-style:italic">Algebra: Gruppen – Ringe – Körper</cite>. <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>181</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Nullfunktion&rft.au=Karpfinger%2C+Meyberg&rft.btitle=Algebra%3A+Gruppen+-+Ringe+-+K%C3%B6rper&rft.genre=book&rft.pages=181" style="display:none"> </span></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a></span> <span class="reference-text">Karpfinger, Meyberg: <cite style="font-style:italic">Algebra: Gruppen – Ringe – Körper</cite>. <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>172</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Nullfunktion&rft.au=Karpfinger%2C+Meyberg&rft.btitle=Algebra%3A+Gruppen+-+Ringe+-+K%C3%B6rper&rft.genre=book&rft.pages=172" style="display:none"> </span></span>

</li>

</ol>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<ul><li><a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ZeroMap.html"><i>Zero Map</i>.</a> In: <i><a href="MathWorld" title="MathWorld">MathWorld</a></i> (englisch).</li>

<li>matte, yark: <a rel="nofollow" class="external text" href="https://planetmath.org/ZeroMap"><i>Zero Map</i>.</a> In: <i><a href="PlanetMath" title="PlanetMath">PlanetMath</a>.</i> (englisch)</li></ul></div><div><div class="zim-footer">

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