Schwach-*-Topologie

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Die Bedeutung beruht u. a. auf dem <a href="/wiki/Satz_von_Banach-Alaoglu" title="Satz von Banach-Alaoglu">Satz von Banach-Alaoglu</a>, wonach die <a href="/wiki/Einheitskugel" title="Einheitskugel">Einheitskugel</a> im Dualraum bezüglich dieser Topologie <a href="/wiki/Kompakter_Raum" title="Kompakter Raum">kompakt</a> ist. Die schwach-*-Topologie spielt eine wichtige Rolle in vielen <a href="/wiki/Funktionalanalysis" title="Funktionalanalysis">funktionalanalytischen</a> Konstruktionen, so zum Beispiel in der <a href="/wiki/Gelfand-Transformation" title="Gelfand-Transformation">Gelfand-Transformation</a> oder im <a href="/wiki/Satz_von_Mackey-Arens" title="Satz von Mackey-Arens">Satz von Mackey-Arens</a>, der diejenigen Topologien auf einem lokalkonvexen Raum beschreibt, die zum selben topologischen Dualraum wie die Ausgangstopologie führen. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Definition"><span class="tocnumber">1</span> <span class="toctext">Definition</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Konvergenz"><span class="tocnumber">2</span> <span class="toctext">Konvergenz</span></a></li> <li class="toclevel-1 tocsection-3"><a href="#Halbnormen"><span class="tocnumber">3</span> <span class="toctext">Halbnormen</span></a></li> <li class="toclevel-1 tocsection-4"><a href="#Produkttopologie"><span class="tocnumber">4</span> <span class="toctext">Produkttopologie</span></a></li> <li class="toclevel-1 tocsection-5"><a href="#Eigenschaften"><span class="tocnumber">5</span> <span class="toctext">Eigenschaften</span></a></li> <li class="toclevel-1 tocsection-6"><a href="#Siehe_auch"><span class="tocnumber">6</span> <span class="toctext">Siehe auch</span></a></li> <li class="toclevel-1 tocsection-7"><a href="#Literatur"><span class="tocnumber">7</span> <span class="toctext">Literatur</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schwach-*-Topologie&veaction=edit&section=1" title="Abschnitt bearbeiten: Definition" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Schwach-*-Topologie&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Definition"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Jedes Element <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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 x}"></span> aus einem normierten oder allgemeiner lokalkonvexen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }"></span>-<a href="/wiki/Vektorraum" title="Vektorraum">Vektorraum</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {K} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {K} }"></span> ist hier <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> oder <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>) definiert durch die Formel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {x}}(f):=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {x}}(f):=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9164d57a48941492a9cd7c9ced2fe88835d729b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.58ex; height:2.843ex;" alt="{\displaystyle {\hat {x}}(f):=f(x)}"></span> ein <a href="/wiki/Funktional" title="Funktional">lineares Funktional</a> auf dem topologischen Dualraum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a574600572696493d48300245a45b8de0638ce21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle E'}"></span>. Die schwach-*-Topologie ist definiert als die <a href="/wiki/Initialtopologie" title="Initialtopologie">schwächste Topologie</a> auf <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a574600572696493d48300245a45b8de0638ce21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle E'}"></span>, die all diese Abbildungen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {x}}\colon E'\rightarrow \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>:</mo> <msup> <mi>E</mi> <mo>′</mo> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {x}}\colon E'\rightarrow \mathbb {K} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b92cc299d252f7ed0db120f594a10542417669a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.264ex; height:2.509ex;" alt="{\displaystyle {\hat {x}}\colon E'\rightarrow \mathbb {K} }"></span> <a href="/wiki/Stetige_Funktion" title="Stetige Funktion">stetig</a> macht. </p><p>Eine etwas konkretere Definition erhält man durch die Angabe einer <a href="/wiki/Umgebungsbasis" title="Umgebungsbasis">Umgebungsbasis</a>. Für <span class="mwe-math-element"><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\in E'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <msup> <mi>E</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in E'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7deaae0714d0bd8baa51e9d529997b04956dab6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.598ex; height:2.843ex;" alt="{\displaystyle f\in E'}"></span> bilden die Mengen </p> <dl><dd><span class="mwe-math-element"><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_{f}(x_{1},\ldots ,x_{n},\epsilon ):=\{g\in E'\,\vert \,\forall j=1,\ldots ,n\colon |f(x_{j})-g(x_{j})|<\epsilon \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>g</mi> <mo>∈</mo> <msup> <mi>E</mi> <mo>′</mo> </msup> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">|</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">∀</mi> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ϵ</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{f}(x_{1},\ldots ,x_{n},\epsilon ):=\{g\in E'\,\vert \,\forall j=1,\ldots ,n\colon |f(x_{j})-g(x_{j})|<\epsilon \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58992c660fa3885ec5130f25328b9b4c4c5a67a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:63.335ex; height:3.176ex;" alt="{\displaystyle U_{f}(x_{1},\ldots ,x_{n},\epsilon ):=\{g\in E'\,\vert \,\forall j=1,\ldots ,n\colon |f(x_{j})-g(x_{j})|<\epsilon \}}"></span>,</dd></dl> <p>wobei <span class="mwe-math-element"><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_{1},\ldots ,x_{n}\in E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},\ldots ,x_{n}\in E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d360f8cda229d5ddabd0ae97aefe2a3a4331f59c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.727ex; height:2.509ex;" alt="{\displaystyle x_{1},\ldots ,x_{n}\in E}"></span>, <span class="mwe-math-element"><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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" 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} }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568095ad3924314374a5ab68fae17343661f2a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}"></span>, eine Umgebungsbasis schwach-*-offener Mengen von <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>. Die schwach-*-Topologie wird oft mit w<sup>*</sup> bezeichnet, nach der englischen Bezeichnung weak-*-topology, oder mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (E',E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <msup> <mi>E</mi> <mo>′</mo> </msup> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (E',E)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dfb980019dd2ac2b410eb396da9e8cf4a884fe6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.427ex; height:3.009ex;" alt="{\displaystyle \sigma (E',E)}"></span>, um die Herkunft als Initialtopologie anzudeuten. </p> <div class="mw-heading mw-heading2"><h2 id="Konvergenz">Konvergenz</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schwach-*-Topologie&veaction=edit&section=2" title="Abschnitt bearbeiten: Konvergenz" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Schwach-*-Topologie&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Konvergenz"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Eine <a href="/wiki/Folge_(Mathematik)" title="Folge (Mathematik)">Folge</a> <span class="mwe-math-element"><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_{n})_{n\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f_{n})_{n\in \mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d822f0926e32e090d67100eca20476c144bee03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.668ex; height:2.843ex;" alt="{\displaystyle (f_{n})_{n\in \mathbb {N} }}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/853ec696b96371acbd7aeb56bbc9db327d49e768" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle E^{\prime }}"></span> (oder allgemeiner ein <a href="/wiki/Netz_(Topologie)" title="Netz (Topologie)">Netz</a> <span class="mwe-math-element"><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_{i})_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f_{i})_{i\in I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f3dd9718b5e3460904d390a0d092237aec266ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.473ex; height:2.843ex;" alt="{\displaystyle (f_{i})_{i\in I}}"></span>) konvergiert genau dann in der schwach-*-Topologie gegen <span class="mwe-math-element"><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\in E^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in E^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34a397bd1afe585d1080c839216b11407c60de00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.598ex; height:2.843ex;" alt="{\displaystyle f\in E^{\prime }}"></span>, wenn </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }f_{n}(x)=f(x)\quad {\text{(bzw. }}\lim _{i\in I}f_{i}(x)=f(x){\text{)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>(bzw. </mtext> </mrow> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>)</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }f_{n}(x)=f(x)\quad {\text{(bzw. }}\lim _{i\in I}f_{i}(x)=f(x){\text{)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d00b5d9d707cd1e496811bb0586019bf4c1a78ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:43.632ex; height:4.009ex;" alt="{\displaystyle \lim _{n\to \infty }f_{n}(x)=f(x)\quad {\text{(bzw. }}\lim _{i\in I}f_{i}(x)=f(x){\text{)}}}"></span></dd></dl> <p>für alle <span class="mwe-math-element"><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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30b1971b01bc31d5b816f03cc7e1d9215d6c2ad8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.946ex; height:2.176ex;" alt="{\displaystyle x\in E}"></span> gilt. Daher nennt man die schwach-*-Topologie auch die <i>Topologie der punktweisen Konvergenz</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Halbnormen">Halbnormen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schwach-*-Topologie&veaction=edit&section=3" title="Abschnitt bearbeiten: Halbnormen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Schwach-*-Topologie&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Halbnormen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Der Dualraum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a574600572696493d48300245a45b8de0638ce21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle E'}"></span> ist mit der Schwach-*-Topologie ein lokalkonvexer Raum. Die schwach-*-Topologie kann daher auch durch die Angabe eines <a href="/wiki/Halbnorm" title="Halbnorm">Halbnorm</a>ensystems definiert werden. Mit <span class="mwe-math-element"><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_{1},\ldots ,x_{n}\in E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},\ldots ,x_{n}\in E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d360f8cda229d5ddabd0ae97aefe2a3a4331f59c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.727ex; height:2.509ex;" alt="{\displaystyle x_{1},\ldots ,x_{n}\in E}"></span>, <span class="mwe-math-element"><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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" 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} }"></span>, bilden die Halbnormen </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{x_{1},\ldots ,x_{n}}(f):=\max\{|f(x_{1})|,\ldots ,|f(x_{n})|\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{x_{1},\ldots ,x_{n}}(f):=\max\{|f(x_{1})|,\ldots ,|f(x_{n})|\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d38f17cb18a6f8f26dcaf35a6ef6acce541b871e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:40.366ex; height:3.009ex;" alt="{\displaystyle p_{x_{1},\ldots ,x_{n}}(f):=\max\{|f(x_{1})|,\ldots ,|f(x_{n})|\}}"></span></dd></dl> <p>ein solches System. </p> <div class="mw-heading mw-heading2"><h2 id="Produkttopologie">Produkttopologie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schwach-*-Topologie&veaction=edit&section=4" title="Abschnitt bearbeiten: Produkttopologie" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Schwach-*-Topologie&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Produkttopologie"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Es gilt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle E'\subset \prod _{x\in E}\mathbb {K} =\mathbb {K} ^{E}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>E</mi> <mo>′</mo> </msup> <mo>⊂</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>E</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle E'\subset \prod _{x\in E}\mathbb {K} =\mathbb {K} ^{E}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4f303ab35e9f6164f969e3a0b259d5fd43cb208" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.885ex; height:3.343ex;" alt="{\displaystyle \textstyle E'\subset \prod _{x\in E}\mathbb {K} =\mathbb {K} ^{E}}"></span>, denn das <a href="/wiki/Kartesisches_Produkt" title="Kartesisches Produkt">kartesische Produkt</a> auf der rechten Seite ist nichts anderes als die Menge aller Funktionen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\rightarrow \mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\rightarrow \mathbb {K} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2088c3518f6a39ad81b52ba74af62268b108b95f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.198ex; height:2.176ex;" alt="{\displaystyle E\rightarrow \mathbb {K} }"></span>. Da die schwach-*-Topologie, wie oben beschrieben, die Topologie der punktweisen Konvergenz ist, kann man diese auch als <a href="/wiki/Relativtopologie" class="mw-redirect" title="Relativtopologie">Relativtopologie</a> der <a href="/wiki/Produkttopologie" title="Produkttopologie">Produkttopologie</a> auf <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \prod _{x\in E}\mathbb {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>E</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \prod _{x\in E}\mathbb {K} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/774b2c61a6a574491e1890c7b5b5fef9a48c25fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.913ex; height:3.009ex;" alt="{\displaystyle \textstyle \prod _{x\in E}\mathbb {K} }"></span> beschreiben. </p><p>Im Produktraum ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \prod _{x\in E}\{\lambda \in {\mathbb {K} }\,\vert \,|\lambda |\leq r_{\lambda }\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>E</mi> </mrow> </munder> <mo fence="false" stretchy="false">{</mo> <mi>λ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">|</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \prod _{x\in E}\{\lambda \in {\mathbb {K} }\,\vert \,|\lambda |\leq r_{\lambda }\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4819de49828576c19998fbb795ea7ee20b954bcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.455ex; height:3.009ex;" alt="{\displaystyle \textstyle \prod _{x\in E}\{\lambda \in {\mathbb {K} }\,\vert \,|\lambda |\leq r_{\lambda }\}}"></span> nach dem <a href="/wiki/Satz_von_Tychonoff" title="Satz von Tychonoff">Satz von Tychonoff</a> für jede Wahl positiver reeller Zahlen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\lambda }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\lambda }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b202fb3bb01836f93a38ebba25549d037da5226f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.239ex; height:2.009ex;" alt="{\displaystyle r_{\lambda }}"></span> eine <a href="/wiki/Kompakter_Raum" title="Kompakter Raum">kompakte</a> Untermenge. Diese Tatsache ist ein wesentlicher Schritt im Beweis des <a href="/wiki/Satz_von_Banach-Alaoglu" title="Satz von Banach-Alaoglu">Satzes von Banach-Alaoglu</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schwach-*-Topologie&veaction=edit&section=5" title="Abschnitt bearbeiten: Eigenschaften" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Schwach-*-Topologie&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Eigenschaften"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Die schwach-*-Topologie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (E',E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <msup> <mi>E</mi> <mo>′</mo> </msup> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (E',E)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dfb980019dd2ac2b410eb396da9e8cf4a884fe6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.427ex; height:3.009ex;" alt="{\displaystyle \sigma (E',E)}"></span> macht <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a574600572696493d48300245a45b8de0638ce21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle E'}"></span> zu einem <a href="/wiki/Lokalkonvexer_Raum" title="Lokalkonvexer Raum">lokalkonvexen Raum</a>. Bildet man bezüglich dieser Topologie den <a href="/wiki/Dualraum#Starker_Dualraum_eines_lokalkonvexen_Raums" title="Dualraum">starken Dualraum</a>, so erhält man <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\hat {x}}\,\vert \,x\in E\}\cong E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">|</mo> <mspace width="thinmathspace" /> <mi>x</mi> <mo>∈</mo> <mi>E</mi> <mo fence="false" stretchy="false">}</mo> <mo>≅</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{{\hat {x}}\,\vert \,x\in E\}\cong E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcb5c6fea0fef2e9155b825a4279fe6bba799564" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.896ex; height:2.843ex;" alt="{\displaystyle \{{\hat {x}}\,\vert \,x\in E\}\cong E}"></span>, oder kurz</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E',\sigma (E',E))'\cong E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>E</mi> <mo>′</mo> </msup> <mo>,</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <msup> <mi>E</mi> <mo>′</mo> </msup> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo>≅</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E',\sigma (E',E))'\cong E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc7833f468109df6c2d2d8654407e708ec4bca4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.308ex; height:3.009ex;" alt="{\displaystyle (E',\sigma (E',E))'\cong E}"></span>.</dd></dl></dd></dl> <ul><li>Die wohl wichtigste Eigenschaft im Fall normierter Räume wird im <a href="/wiki/Satz_von_Banach-Alaoglu" title="Satz von Banach-Alaoglu">Satz von Banach-Alaoglu</a> behandelt, das ist die schwach-*-Kompaktheit der Einheitskugel im Dualraum.</li> <li>Durch die kanonische Einbettung eines Banachraums <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> in seinen <a href="/wiki/Bidualraum" class="mw-redirect" title="Bidualraum">Bidualraum</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0ae2240bc3213fc7bc6e68a6dfda7d89ae34f13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.931ex; height:2.509ex;" alt="{\displaystyle E''}"></span> kann man <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> als Unterraum von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0ae2240bc3213fc7bc6e68a6dfda7d89ae34f13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.931ex; height:2.509ex;" alt="{\displaystyle E''}"></span> ansehen. Der <a href="/wiki/Satz_von_Hahn-Banach" title="Satz von Hahn-Banach">Satz von Hahn-Banach</a> zeigt, dass <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> bezüglich der schwach-*-Topologie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (E'',E')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <msup> <mi>E</mi> <mo>″</mo> </msup> <mo>,</mo> <msup> <mi>E</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (E'',E')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d08a5c2cf8fa68a15be3bc51ea99e185c1c0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.583ex; height:3.009ex;" alt="{\displaystyle \sigma (E'',E')}"></span> <a href="/wiki/Dichte_Teilmenge" title="Dichte Teilmenge">dicht</a> liegt. Mit Hilfe des <a href="/wiki/Trennungssatz" title="Trennungssatz">Trennungssatzes</a> kann man zeigen, dass diese Dichtheitsbeziehung bei normierten Räumen auch für die <a href="/wiki/Einheitskugel" title="Einheitskugel">Einheitskugeln</a> richtig ist, das heißt, es gilt der auf <a href="/wiki/Herman_H._Goldstine" title="Herman H. Goldstine">Herman H. Goldstine</a> zurückgehende</li></ul> <dl><dd><dl><dd><i>Satz von Goldstine:</i> <span class="mwe-math-element"><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 E\,\vert \,\|x\|\leq 1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>E</mi> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">|</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x\in E\,\vert \,\|x\|\leq 1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bcbfbeb25513819998fd01569e148ccda047a80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.607ex; height:2.843ex;" alt="{\displaystyle \{x\in E\,\vert \,\|x\|\leq 1\}}"></span> liegt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (E'',E')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <msup> <mi>E</mi> <mo>″</mo> </msup> <mo>,</mo> <msup> <mi>E</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (E'',E')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d08a5c2cf8fa68a15be3bc51ea99e185c1c0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.583ex; height:3.009ex;" alt="{\displaystyle \sigma (E'',E')}"></span>-dicht in <span class="mwe-math-element"><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 E''\,\vert \,\|x''\|\leq 1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msup> <mi>x</mi> <mo>″</mo> </msup> <mo>∈</mo> <msup> <mi>E</mi> <mo>″</mo> </msup> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">|</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">‖</mo> <msup> <mi>x</mi> <mo>″</mo> </msup> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x''\in E''\,\vert \,\|x''\|\leq 1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36fc341366536f577163ae5a157083848f07686d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.037ex; height:3.009ex;" alt="{\displaystyle \{x''\in E''\,\vert \,\|x''\|\leq 1\}}"></span>.</dd></dl></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schwach-*-Topologie&veaction=edit&section=6" title="Abschnitt bearbeiten: Siehe auch" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Schwach-*-Topologie&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Siehe auch"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Beschr%C3%A4nkte_schwach-*-Topologie" title="Beschränkte schwach-*-Topologie">Beschränkte schwach-*-Topologie</a></li> <li><a href="/wiki/Initialtopologie" title="Initialtopologie">Initialtopologie</a></li> <li><a href="/wiki/Produkttopologie" title="Produkttopologie">Produkttopologie</a></li> <li><a href="/wiki/Schwache_Topologie" title="Schwache Topologie">Schwache Topologie</a></li> <li><a href="/wiki/Starke_Operatortopologie" class="mw-redirect" title="Starke Operatortopologie">Starke Operatortopologie</a></li> <li><a href="/wiki/Ultraschwache_Topologie" class="mw-redirect" title="Ultraschwache Topologie">Ultraschwache Topologie</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schwach-*-Topologie&veaction=edit&section=7" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Schwach-*-Topologie&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Literatur"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Klaus Floret, Joseph Wloka: <i>Einführung in die Theorie der lokalkonvexen Räume</i> (= <i>Lecture Notes in Mathematics.</i> Bd. 56, <span class="plainlinks-print"><a href="/wiki/Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a> <span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%220075-8434%22&key=cql">0075-8434</a></span></span>). Springer, Berlin u. a. 1968.</li></ul></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Abgerufen von „<a dir="ltr" href="https://de.wikipedia.org/w/index.php?title=Schwach-*-Topologie&oldid=248303432#Eigenschaften">https://de.wikipedia.org/w/index.php?title=Schwach-*-Topologie&oldid=248303432#Eigenschaften</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorien" title="Wikipedia:Kategorien">Kategorien</a>: <ul><li><a href="/wiki/Kategorie:Topologische_Struktur" title="Kategorie:Topologische Struktur">Topologische Struktur</a></li><li><a href="/wiki/Kategorie:Funktionalanalysis" title="Kategorie:Funktionalanalysis">Funktionalanalysis</a></li><li><a href="/wiki/Kategorie:Mengentheoretische_Topologie" title="Kategorie:Mengentheoretische Topologie">Mengentheoretische Topologie</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigationsmenü</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Meine Werkzeuge</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item"><span title="Benutzerseite der IP-Adresse, von der aus du Änderungen durchführst">Nicht angemeldet</span></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Spezial:Meine_Diskussionsseite" title="Diskussion über Änderungen von dieser IP-Adresse [n]" accesskey="n"><span>Diskussionsseite</span></a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Spezial:Meine_Beitr%C3%A4ge" title="Eine Liste der Bearbeitungen, die von dieser IP-Adresse gemacht wurden [y]" accesskey="y"><span>Beiträge</span></a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Spezial:Benutzerkonto_anlegen&returnto=Schwach-%2A-Topologie" title="Wir ermutigen dich dazu, ein Benutzerkonto zu erstellen und dich anzumelden. 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Schwach-*-Topologie – Wikipedia