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01FA2:Kapitola5 - Historie editací
2024-03-29T15:36:25Z
Historie editací této stránky
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https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01FA2:Kapitola5&diff=8224&oldid=prev
Kubuondr v 6. 2. 2019, 08:05
2019-02-06T08:05:07Z
<p></p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr style='vertical-align: top;'>
<td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td>
<td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 6. 2. 2019, 08:05</td>
</tr><tr><td colspan="2" class="diff-lineno" id="L208" >Řádka 208:</td>
<td colspan="2" class="diff-lineno">Řádka 208:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>       B)^*$.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>       B)^*$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>       \[\Gamma(B^*)=U(\Gamma(B))^\perp=(\uz{U(\Gamma(B))})^\perp=</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>       \[\Gamma(B^*)=U(\Gamma(B))^\perp=(\uz{U(\Gamma(B))})^\perp=</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>       (U(\uz{\Gamma(B)}))^\perp=U(\Gamma(\uz B))=\Gamma((\uz B)^*).\]</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>       (U(\uz{\Gamma(B)}))^\perp=U(\Gamma(\uz B))<ins class="diffchange diffchange-inline">^\perp</ins>=\Gamma((\uz B)^*).\]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>       Druhá rovnost zleva plyne ze spojitosti skalárního součinu, třetí</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>       Druhá rovnost zleva plyne ze spojitosti skalárního součinu, třetí</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>       z~unitarity $U$ a čtvrtá z~uzavíratelnosti $B$.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>       z~unitarity $U$ a čtvrtá z~uzavíratelnosti $B$.</div></td></tr>
</table>
Kubuondr
https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01FA2:Kapitola5&diff=8223&oldid=prev
Kubuondr v 5. 2. 2019, 21:22
2019-02-05T21:22:46Z
<p></p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr style='vertical-align: top;'>
<td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td>
<td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 5. 2. 2019, 21:22</td>
</tr><tr><td colspan="2" class="diff-lineno" id="L134" >Řádka 134:</td>
<td colspan="2" class="diff-lineno">Řádka 134:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>       [x,y]\in\Gamma(A^*)&\iff\forall u\in\Dom A:(x,Au)=(y,u)\\</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>       [x,y]\in\Gamma(A^*)&\iff\forall u\in\Dom A:(x,Au)=(y,u)\\</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>       &\iff\forall[u,Au]\in\Gamma(A):</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>       &\iff\forall[u,Au]\in\Gamma(A):</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>       ([x,y],\underbrace{[Au,-u]}_{U[u,Au]})\\</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>       ([x,y],\underbrace{[Au,-u]}_{U[u,Au]})<ins class="diffchange diffchange-inline">=0</ins>\\</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>       &\iff\forall[u,v]\in\Gamma(A):([x,y],U[u,v])=0\\</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>       &\iff\forall[u,v]\in\Gamma(A):([x,y],U[u,v])=0\\</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>       &\iff[x,y]\in U(\Gamma(A))^\perp.\qed</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>       &\iff[x,y]\in U(\Gamma(A))^\perp.\qed</div></td></tr>
</table>
Kubuondr
https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=01FA2:Kapitola5&diff=3109&oldid=prev
Admin: Založena nová stránka: %\wikiskriptum{01FA2} \section{Neomezené operátory} Obecně $\Dom A\not=\H$, obvykle požadujeme $\uz{\Dom A}=\H$. \begin{enumerate} \item $A=B\iff\Dom A=\Dom B\wedge...
2010-07-31T23:30:10Z
<p>Založena nová stránka: %\wikiskriptum{01FA2} \section{Neomezené operátory} Obecně $\Dom A\not=\H$, obvykle požadujeme $\uz{\Dom A}=\H$. \begin{enumerate} \item $A=B\iff\Dom A=\Dom B\wedge...</p>
<p><b>Nová stránka</b></p><div>%\wikiskriptum{01FA2}<br />
\section{Neomezené operátory}<br />
<br />
Obecně $\Dom A\not=\H$, obvykle požadujeme $\uz{\Dom A}=\H$.<br />
<br />
\begin{enumerate}<br />
\item $A=B\iff\Dom A=\Dom B\wedge Ax=Bx\ \forall x\in\Dom A$,<br />
\item $C=AB$: $\Dom C=\{x\in\Dom B|Bx\in\Dom A\}$, $Cx=ABx\ \forall<br />
x\in\Dom C$,<br />
\item $C=A+B$: $\Dom C=\Dom A\cap\Dom B$.<br />
\end{enumerate}<br />
<br />
\begin{define}<br />
Nechť $\uz{\Dom A}=\H$. Potom $x\in \Dom A^*$, právě když existuje<br />
$u\in\H$ tak, že pro každé $y\in\Dom A$ je $(x,Ay)=(u,y)$. Jestliže<br />
$u\in\H$ existuje, pak je určeno jednoznačně: Kdyby existovalo<br />
$u'\in\H$ tak, že pro každé $y\in\Dom A$ $(x,Ay)=(u',y)$, pak pro<br />
každé $y\in\Dom A$ je $(u-u',y)=0$ a $u-u'\in\Dom A^\perp=(\uz{\Dom<br />
A})^\perp=\{0\}$. Pokládáme $A^*x=u$.<br />
\end{define}<br />
<br />
\begin{tvrzeni}<br />
Platí:<br />
\begin{enumerate}[(i)]<br />
\item $(\lambda A)^*=\uz{\lambda}A^*$ ($\lambda\not=0$);<br />
\item $(A+B)^*\supset A^*+B^*$, pokud je levá strana definovaná<br />
($\uz{\Dom (A+B)}=\uz{\Dom A\cap \Dom B}=\H$);<br />
\item $(AB)^*\supset B^*A^*$, pokud je levá strana definovaná<br />
($\Dom AB=\H$);<br />
\item $A\subset B\implies B^*\subset A^*$.<br />
\end{enumerate}<br />
\begin{proof}<br />
\begin{enumerate}<br />
\item Buď $x\in\Dom A^*=\Dom(\lambda A^*)$ (platí pro<br />
$\lambda\not=0$). Pro $y\in\Dom(\lambda A)$ platí<br />
\[(x,(\lambda A)y)=(x,\lambda(Ay))=\lambda(x,Ay)=\lambda(A^*x,y)=<br />
(\overline\lambda A^*x,y),\] <br />
z~čehož plyne $x\in\Dom (\lambda A)^*$, díky jednoznačnosti<br />
$\overline\lambda A^*x=(\lambda A)^*x$ a $\overline\lambda<br />
A^*\subset (\lambda A)^*$.<br />
<br />
Naopak buď $x\in\Dom(\lambda A)^*$, pak<br />
\[(\overline\lambda x,Ay)=(x,\lambda Ay)=((\lambda A)^*x,y).\]<br />
Proto $x\in\Dom(\overline\lambda A^*)$ a $(\lambda<br />
A)^*\subset\overline\lambda A^*$. Celkem<br />
$(\lambda A)^*=\overline\lambda A^*$.<br />
\item Buď $x\in\Dom(A^*+B^*)$, tedy $x\in\Dom A^*\cap\Dom B^*$,<br />
$y\in\Dom(A+B)$. Pak<br />
\[(x,(A+B)y)=(x,Ay)+(x,By)=(A^*x,y)+(B^*x,y)=((A^*+B^*)x,y).\]<br />
\item Buď $x\in\Dom(B^*A^*)\iff x\in\Dom B^*\wedge A^*x\in\Dom<br />
B^*$. Pro každé $y\in\Dom(AB)$, tj. $y\in\Dom B$, $By\in\Dom A$<br />
platí<br />
\[(x,(AB)y)=(x,A(By))=(A^*x,By)=(B^*(A^*x),y)=(B^*A^*x,y).\]<br />
Proto $x\in\Dom(AB)^*$, $(AB)^*x=(B^*A^*)x$ a $B^*A^*\subset<br />
(AB)^*$.<br />
\item Buď $x\in\Dom B^*$. Pro $y\in\Dom(A)\subset\Dom(B)$ platí<br />
\[(x,Ay)=(x,By)=(B^*x,y)\]<br />
a $x\in\Dom A^*$, $A^*x=B^*x\implies B^*\subset A^*$.\qed<br />
\end{enumerate}<br />
\noqed<br />
\end{proof}<br />
\end{tvrzeni}<br />
<br />
\begin{theorem}<br />
Nechť $\uz{\Dom A}=\H$, $A^{-1}$ existuje ($\Ker A=0$) a<br />
$\uz{\Dom A^{-1}}=\H$ ($\Dom A^{-1}=\Ran A$). Potom $(A^*)^{-1}$<br />
existuje a platí $(A^*)^{-1}=(A^{-1})^*$.<br />
\begin{proof}<br />
\begin{enumerate}<br />
\item Buď $x\in\Ker A^*$. Potom pro každé $y\in\Dom A$ je<br />
$(x,Ay)=(0,y)=0$. Protože $Ay\subset\Ran A=\Dom A^{-1}$, který<br />
je hustý, je $x=0$, tj. $\Ker A^*=\{0\}$ a tedy $(A^*)^{-1}$<br />
existuje.<br />
\item Buď $y\in\Dom(A^{-1})=\Ran A$, $y=Au$, $u\in\Dom A$,<br />
$(x,A^{-1}y)=(x,u)$.<br />
<br />
Pro každé $u\in\Dom A$, $x\in\Dom(A^*)^{-1}$ je<br />
\[(x,A^{-1}Au)=(x,u)=(A^*(A^*)^{-1}x,u)=((A^*)^{-1}x,Au).\]<br />
Pro $y\in\Dom(A^{-1})$, $x\in\Dom(A^*)^{-1}$ je<br />
$(x,A^{-1}y)=((A^*)^{-1}x,y)$, tudíž $x\in\Dom(A^{-1})^*$ a<br />
$(A^{-1})^*x=(A^*)^{-1}x$.<br />
\item Buď $y\in\Dom(A^{-1})^*$, $x\in\Dom A^{-1}$. Potom<br />
\[((A^{-1})^*y,AA^{-1}x)=((A^{-1})^*y,x)=(y,A^{-1}x).\]<br />
Pro $y\in\Dom(A^{-1})^*$, $z\in\Dom A$ je<br />
$((A^{-1})^*y,Az)=(y,z)$. Proto $(A^{-1})^*y\in\Dom A^*$,<br />
$A^*(A^{-1})^*y=y\in\Ran A^*=\Dom(A^*)^{-1}$,<br />
$(A^*)^{-1}y=(A^{-1})^*y$ a $(A^{-1})^*\subset(A^*)^{-1}$.<br />
Celkem $(A^{-1})^*=(A^*)^{-1}$.\qed<br />
\end{enumerate}<br />
\noqed<br />
\end{proof}<br />
\end{theorem}<br />
<br />
\begin{remark}<br />
\begin{enumerate}<br />
\item $A=\uz A\implies\uz{\Ker A}=\Ker A$.<br />
\item Je-li $\uz{\Dom A}=\H$, potom $\Ran(A-\lambda<br />
I)^\perp=\Ker(A^*-\overline\lambda I)$.<br />
\begin{proof}<br />
$x\in\Ran(A-\lambda I)^\perp\iff(x,(A-\lambda I)y)=0\ \forall<br />
y\in\Dom A\iff x\in\Ker(A^*-\overline\lambda I)$.<br />
\end{proof}<br />
\item Jestliže $\uz{\Dom A}=\H$, $B\in\B(\H)$, pak $(A+B)^*=A^*+B^*$<br />
a $\Dom(A+B)=\Dom(A)$. Specielně $(A-\lambda<br />
I)^*=A^*-\overline\lambda I$.<br />
\item Je-li $A\in\B(\H)$, pak $A^{**}=A$. Je-li $A$ neomezený,<br />
potom $A^{**}$ existuje, právě když $\uz{\Dom A^*}=\H$; potom<br />
$A^{**}=\uz A$.<br />
\item $\Gamma(\uz A)=\uz{\Gamma(A)}$,<br />
$\Gamma(A)\subset\H\oplus\H$. Definujeme<br />
$([x,y],[x',y'])=(x,x')+(y,y')$,<br />
$\norm{[x,y]}=\sqrt{\norm{x}^2+\norm{y}^2}$. Označme<br />
$U:\H\oplus\H\mapsto\H\oplus\H:[x,y]\mapsto[y,-x]$. Zřejmě<br />
$U^2=-I$, $U^*=U^{-1}=-U$.<br />
\item Buď $M\subset \H\oplus\H$. Pak $U(M)^\perp=U(M^\perp)$:<br />
\[\begin{split}<br />
[x,y]\in U(M)^\perp&\iff\forall[u,v]\in U(\Gamma):([x,y],[u,v])=0\\<br />
&\iff\forall[u,v]\in\Gamma:([x,y],[v,-u])=0\\<br />
&\iff(x,v)-(y,u)=0,<br />
\end{split}\]<br />
\[\begin{split}<br />
[x,y]\in U(M^\perp)&\iff[-y,x]\in\Gamma^\perp\\<br />
&\iff\forall[u,v]\in M([-y,x],[u,v])=0\\<br />
&\iff -(y,u)+(x,v)=0.<br />
\end{split}\]<br />
\end{enumerate}<br />
\end{remark}<br />
<br />
\begin{lemma}<br />
Nechť $\uz{\Dom A}=\H$. Potom $\Gamma(A^*)=U(\Gamma(A))^\perp$.<br />
\begin{proof}<br />
\[<br />
\begin{split}<br />
[x,y]\in\Gamma(A^*)&\iff\forall u\in\Dom A:(x,Au)=(y,u)\\<br />
&\iff\forall[u,Au]\in\Gamma(A):<br />
([x,y],\underbrace{[Au,-u]}_{U[u,Au]})\\<br />
&\iff\forall[u,v]\in\Gamma(A):([x,y],U[u,v])=0\\<br />
&\iff[x,y]\in U(\Gamma(A))^\perp.\qed<br />
\end{split}<br />
\]<br />
\noqed<br />
\end{proof}<br />
\end{lemma}<br />
<br />
\begin{dusl}<br />
$A^*$ je uzavřený, neboť $\Gamma(A^*)=\uz{\Gamma(A^*)}$.<br />
\end{dusl}<br />
<br />
\begin{theorem}<br />
Nechť $\uz{\Dom A}=\H$. Potom $A^{**}=(A^*)^*$ existuje, právě když<br />
$A$ je uzavíratelný a navíc $A^{**}=\uz{A}$.<br />
\begin{proof}<br />
$A^{**}$ existuje $\iff\uz{\Dom<br />
A^*}=\H\iff\Dom(A^*)^\perp=\{0\}$. Dále<br />
\[\begin{split}<br />
[x,0]\in\Gamma(A^*)^\perp&\iff<br />
\forall[u,v]\in\Gamma(A^*):0=([x,0],[u,v])=(x,u)\\<br />
&\iff x\in\Dom(A^*)^\perp<br />
\end{split}\]<br />
a<br />
\[\begin{split}<br />
x\in\Dom(A^*)^\perp&\iff[x,0]\in\Gamma(A^*)^\perp=<br />
{U(\Gamma(A))^\perp}^\perp=\uz{U(\Gamma(A))}=U(\uz{\Gamma(A)})\\<br />
&\iff -U[x,0]\in\uz{\Gamma(A)}\\&\iff[0,x]\in\uz{\Gamma(A)}.<br />
\end{split}\]<br />
Z~toho plyne<br />
\[\begin{split}<br />
\exists A^{**}&\iff\uz{\Dom A^*}=\H\iff\Dom(A^*)^\perp=\{0\}\\<br />
&\iff\{x\in\H|[0,x]\in\uz{\Gamma(A)}\}=\{0\}\\<br />
&\iff\uz{\Gamma(A)}\text{ je graf}\\<br />
&\iff A\text{ je uzavíratelný}.<br />
\end{split}\]<br />
Konečně<br />
\[\Gamma(A^{**})=U(\Gamma(A^*))^\perp=U(\Gamma(A^*)^\perp)=<br />
U(\uz{U(\Gamma(A))})=U^2(\uz{\Gamma(A)})=<br />
\uz{\Gamma(A)}=\Gamma(\uz{A}).\]<br />
Přitom jsme využili toho, že $\uz{\Gamma(A)}$ je podprostor, takže<br />
$(-1)\uz{\Gamma(A)}=\uz{\Gamma(A)}$.<br />
\end{proof}<br />
\end{theorem}<br />
<br />
\begin{define}<br />
Nechť $A$ je hustě definovaný. Potom<br />
\begin{enumerate}<br />
\item $A$ je symetrický, právě když (ekvivalentní formulace)<br />
\begin{enumerate}<br />
\item $(\forall x,y\in\Dom A)((Ax,y)=(x,Ay))$,<br />
\item $(\forall x\in\Dom A)(x\in\Dom A^*,\ A^*x=Ax)$,<br />
\item $A\subset A^*$.<br />
\end{enumerate}<br />
\item $A$ je samosdružený, právě když $A^*=A$.<br />
\item $A$ je normální, právě když $A^*A=AA^*$ (včetně definičních<br />
oborů).<br />
\end{enumerate}<br />
\end{define}<br />
<br />
\begin{theorem}<br />
\begin{enumerate}<br />
\item Symetrický operátor je uzavíratelný.<br />
\item Uzávěr symetrického operátoru je symetrický.<br />
\item Je-li $A$ symetrický a $\Dom A =\H$, potom $A$ je omezený.<br />
\end{enumerate}<br />
\begin{proof}<br />
\begin{enumerate}<br />
\item $A\subset A^*$, $A^*$ je uzavřený.<br />
\item $A\subset A^*\implies \uz{A}\subset A^*=(\uz A)^*$. Obecně<br />
pro každý $B:\uz{\Dom B}=\H$, uzavíratelný, platí $B^*=(\uz<br />
B)^*$.<br />
\[\Gamma(B^*)=U(\Gamma(B))^\perp=(\uz{U(\Gamma(B))})^\perp=<br />
(U(\uz{\Gamma(B)}))^\perp=U(\Gamma(\uz B))=\Gamma((\uz B)^*).\]<br />
Druhá rovnost zleva plyne ze spojitosti skalárního součinu, třetí<br />
z~unitarity $U$ a čtvrtá z~uzavíratelnosti $B$.<br />
\item $A\subset\uz A$ existuje, $\Dom A=\H$, takže $A=\uz A$ a $A$<br />
je omezený.\qed<br />
\end{enumerate}<br />
\noqed<br />
\end{proof}<br />
\end{theorem}</div>
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