<!1,2018080717:23:31,V.2>
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<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">	\chapter[مشخص‌سازی هم‌ریختی‌های ژوردان بر جبرهای باناخ]{\huge مشخص‌سازی هم‌ریختی‌های ژوردان بر جبرهای باناخ}</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">در این فصل هم‌ریختی‌های ژوردان بین جبرهای باناخ را مورد مطالعه و بررسی قرار می‌دهیم و نشان می‌دهیم تحت شرایطی خاص، هم‌ریختی‌های ژوردان به یک هم‌ریختی تبدیل می‌شوند. </p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">\section{معرفی هم‌ریختی‌های ژوردان}</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">فرض کنیم ‎‏‎‎$\varphi:A\to B$‎ نگاشتی خطی بین جبرهای ‎‎$‎‎A$‎‏ و ‎‎$‎‎B$‎ باشد. نگاشت ‎$\varphi$‎  را هم‌ریختی ژوردان</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">\index{هم‌ریختی ژوردان}</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> نامیم هرگاه </p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\begin{equation}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\varphi(ab+ba)=\varphi(a) \varphi(b)+\varphi(b)\varphi(a) \quad (a,b\in A)‎.</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{equation}‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">یا به‌طور معادل به‌ازای هر ‎$a\in A$‎ ‎،$\varphi(a^2)=\varphi(a)^2$.‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\begin{rem}‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">هر  ‎$2$-‎هم‌ریختی ژوردان را معمولا هم‌ریختی ژوردان گوییم.</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{rem}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">%-----------------------------------------------------------------------------------------------------------</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\begin{dfn}‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">چنانچه ‎$A$‎ و ‎$B$‎ حلقه </p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\emph{(}‎و در حالت خاص جبر باشند‎\emph{)}‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> و نگاشت جمعی  ‎$\varphi$‎ ضربی هم باشد، یعنی </p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎$$\varphi(ab)=\varphi(a)\varphi(b) \quad (a,b\in A),$$‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">نگاشت ‎$\varphi$‎  را هم‌ریختی حلقه‌ای نامیم. برای ‎$n\in \mathbb{N}$‎، درصورتی نگاشت جمعی ‎$\varphi:A\to B$‎ را </p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎$ n$-‎هم‌ریختی حلقه گویند که به‌ازای هر ‎$a_1,a_2,\ldots,a_n \in A$‎، داشته باشیم:</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎$$‎\varphi‎(a_1\cdots a_n)=‎\varphi‎(a_1)\cdots ‎\varphi‎(a_n).$$‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">چنانچه ‎$A$‎ و ‎$B$‎ جبر باشند و ‎$n$-‎هم‌ریختی</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\index{$n$-هم‌ریختی}‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> حلقه خطی هم باشد، پسوند حلقه را به‌کار نمی‌برند و ‎$\varphi$‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">را   ‎$n$-‎هم‌ریختی  نامند.</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{dfn}‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> واضح است که هر هم‌ریختی حقله‌ای، هم‌ریختی ژوردان</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> \index{هم‌ریختی ژوردان}</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">  است ولی عکس این مطلب در حالت کلی برقرار نیست و البته تحت شرایط خاصی برقرار است. برای مثال هر هم‌ریختی ژوردان از جبر باناخ</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">\index{جبر باناخ}</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">  جابه‌جایی ‎$A$‎ به‌توی ‎$\mathbb{C}$‎ یک هم‌ریختی حقله‌‌ ای است. در حالت کلی‌تر زلاسکو در  </p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\cite{1.2}‎ ثابت کرد که هر هم‌ریختی ژوردان از جبر باناخ</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">\index{جبر باناخ}</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> ‎$A$‎ به‏‌توی جبر باناخ جابه‌جایی نیم‌ساده ‎$B$‎ یک هم‌ریختی حلقه‌ای</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">\index{هم‌ریختی حلقه‌ای} </p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">  است </p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\lr{(\cite[Theorem 1.1]{2.2})}‎.</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">در </p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\cite{3.2}‎، اسحاقی گرجی</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\LTRfootnote{Eshaghi Gordji}‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> مفهوم ‎$n$-‎هم‌ریختی‌های ژوردان را برای جبر‌های مختلط مطالعه کرد.</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">وی نشان داد که هر ‎$3$-‎هم‌ریختی ژوردان</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">\index{هم‌ریختی ژوردان}</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> بین جبر‌های باناخ جابه‌جایی ‎$A$‎ و ‎$B$‎ یک ‎$3$-‎هم‌ریختی حلقه‌ای است</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\cite{3.2}.\\‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">علاوه بر آن نشان داد که قضیه‌ی زلاسکو</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\LTRfootnote{Zelazko}‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> برای ‎$3$-‎هم‌ریختی‌های ژوردان نیز‌ برقرار است که به نظر می‌رسد ایرادی در برهان وجود دارد. البته در حالتی که جبر ‎$A$‎ یکدار باشد برهانی دقیق برای این مطلب در </p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\lr{\cite[Theorem 2.4]{Zivari}}</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">ارائه شده‌است.</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">%-----------------------------------------------------------------------------------------------------------</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\begin{rem}‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">در حالت کلی، هسته‌ی یک ‎$n$-‎هم‌ریختی ژوردان</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">\index{هم‌ریختی ژوردان}</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> ممکن است که ایده‌آل نباشد.</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{rem}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">%-----------------------------------------------------------------------------------------------------------</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\begin{exam}‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> فرض کنیم که ‎$A$‎ جبر تمام ماتریس های  ‎$3\times3$‎  بالا مثلثی در ‎$\mathbb{R}$‎ باشد که پایین و روی قطر اصلی آن صفر باشد. در این جبر حاصل‌ضرب هر  ‎$3$‎ عضو دلخواه اعضای آن برابر صفر است. بنابراین هر نگاشت خطی از ‎$A$‎  به ‎$A$‎  یک ‎$3$-‎هم‌ریختی ژوردان است اما هسته‌ی آن لزوما ایده‌آل نیست.</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">%-----------------------------------------------------------------------------------------------------------</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">برای دیدن این مطلب فرض کنیم  </p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎$B=\begin{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0    &amp;a    &amp;b\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0    &amp;0    &amp;c\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0    &amp;0    &amp;0‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{bmatrix}$‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">و</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎$C=\begin{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0    &amp;a'    &amp;b'\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0    &amp;0    &amp;c'\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0    &amp;0    &amp;0‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{bmatrix}$‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">و</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎$D=\begin{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0    &amp;a''    &amp;b''\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0    &amp;0    &amp;c''\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0    &amp;0    &amp;0‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{bmatrix}$‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">اعضای دلخواه جبر ‎$A$‎ باشند. نشان می‌دهیم حاصل‌ضرب این ‎$3$‎ عضو دلخواه صفر است. داریم:</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎$$BC=\begin{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0    &amp;0    &amp;ac'\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0    &amp;0    &amp;0\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0    &amp;0    &amp;0‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎\end{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;"> ‎\Rightarrow  BCD=\begin{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0    &amp;0    &amp;ac'\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0    &amp;0    &amp;0\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0    &amp;0    &amp;0‎   </p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎\end{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎\begin{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0    &amp;a''   &amp;b''\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0    &amp;0    &amp;c''\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0    &amp;0    &amp;0‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎\end{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎=\begin{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0   &amp;0    &amp;0\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0   &amp;0    &amp;0\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0   &amp;0    &amp;0‎ </p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{bmatrix}$$‎ . </p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">حال نشان می‌دهیم نگاشت خطی ‎$\varphi:A\to A$‎ یک ‎$3$-‎هم‌ریختی ژوردان است. از طرفی چون ‎$ \varphi(B) \varphi(C) \varphi(D)=0$‎ باتوجه به تساوی‌های فوق داریم ‎$\varphi(BCD)=\varphi(0)=0$‎. پس ‎$\varphi(BCD)=\varphi(B) \varphi(C) \varphi(D)$‎. درنتیجه ‎$\varphi$‎ یک ‎$3$-‎هم‌ریختی</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">است و لذا ‎$3$-‎هم‌ریختی ژوردان هم هست.</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> حال نگاشت ‎$\varphi:A\to A$‎ را با ضابطه‌ی زیر تعریف می‌کنیم:</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎$$\varphi\left(\begin{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0   &amp;a     &amp;b\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0   &amp;0     &amp;c\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0   &amp;0     &amp;0‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎\end{bmatrix}\right):=\begin{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0   &amp;a+c    &amp;b\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0   &amp;0        &amp;0\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0   &amp;0        &amp;0‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{bmatrix}.$$‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">نشان می‌دهیم نگاشت ‎$\varphi$‎ خطی است ولی هسته‌ی آن ایده‌آل نیست.</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\begin{align*}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\varphi\left(\begin{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;a_1+a_2     &amp;b_1+b_2\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;0       &amp;0\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;0       &amp;0‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{bmatrix}\right)&amp;=\begin{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;a_1+a_2+c_1+c_2     &amp;b_1+b_2\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;0       &amp;0\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;0       &amp;0‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{bmatrix}\\&amp;=\begin{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;a_1+c_1     &amp;b_1\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;0       &amp;0\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;0       &amp;0‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{bmatrix}+\begin{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;a_2+c_2    &amp;b_2\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;0       &amp;0\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;0       &amp;0‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{bmatrix}\\&amp;=\varphi\left(\begin{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;a_1     &amp;b_1\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;0     &amp;c_1\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;0     &amp;0‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{bmatrix}\right)+\varphi\left(\begin{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;a_2     &amp;b_2\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;0     &amp;c_2\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;0     &amp;0‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{bmatrix}\right)‎.</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{align*}‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">ضمناً برای هر عدد حقیقی ‎$\lambda$‎ و هر ماتریس ‎$ B$‎ از جبر ‎$ A$‎، به‌سادگی نشان داده می‌شود که ‎$ \varphi(\lambda B)=\lambda \varphi(B)$‎. از طرفی داریم:</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\begin{align*}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎ker(\varphi)=\Bigg\lbrace \begin{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;a     &amp;b\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;0     &amp;c\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;0     &amp;0‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{bmatrix}‎ :‎\varphi\left(\begin{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;a     &amp;b\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;0     &amp;c\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;0     &amp;0‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{bmatrix}\right)&amp;=\begin{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;a+c    &amp;b\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;0       &amp;0\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;0       &amp;0‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{bmatrix}=\begin{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;0      &amp;0\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;0     &amp;0\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;0     &amp;0‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{bmatrix} \Bigg\rbrace\\ &amp;=\Bigg\lbrace \begin{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;\alpha    &amp;0\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;0           &amp;-\alpha\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎0   &amp;0           &amp;0‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{bmatrix}‎: ‎\alpha\in \mathbb{R}\Bigg\rbrace‎.</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{align*}‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">که نتیجه می‌دهد هسته‌ی ‎$\varphi$‎ ایده‌آل نیست. زیرا اگر فرض کنیم </p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎$\begin{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0    &amp;1    &amp;1\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0    &amp;0    &amp;1\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0    &amp;0    &amp;0‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{bmatrix}\in A$‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">داریم:</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎$$\begin{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0    &amp;1    &amp;1\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0    &amp;0    &amp;1\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0    &amp;0    &amp;0‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎\end{bmatrix} \begin{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0   &amp;\alpha    &amp;0\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0   &amp;0           &amp;-\alpha\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0   &amp;0           &amp;0‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎\end{bmatrix}=\begin{bmatrix}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0   &amp;0    &amp;-\alpha\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0   &amp;0           &amp;0\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎0   &amp;0           &amp;0‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{bmatrix}\notin ker(\varphi).$$‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{exam}‎</p>
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<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\begin{rem}‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">برای ماتریس‌های ‎$ 4\times 4$‎ هم نظیر مثال فوق می‌توان مثالی ارائه کرد که حاصل‌ضرب هر چهار ماتریس بالا مثلثی صفر شود ولی اگر </p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎$\varphi:A\to A$‎ با این ضابطه تعریف شود که هر عضو دلخواه ‎$ A$‎ را به ماتریسی بنگارد که فقط درایه‌ی سطر اول و ستون چهارم آن ماتریس باقی بماند</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">و بقیه‌ی درایه‌های آن صفر یاشند، آنگاه ‎$ ker(\varphi)$‎ ایده‌آل نیست</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\cite{3.2}‎.</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{rem}‎</p>
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<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‏‎... (ادامه مطالب حذف شده است)</p>
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<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">\section{ضرب آرنز}</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">در چند دهه‌ی اخیر ریاضی‌دانان تلاش فراوان کرده‌اند تا بتوانند دوگان دوم یک جبر باناخ</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">\index{جبر باناخ}</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> مانند ‎$A$‎ را به‌عنوان توسیعی از ‎$A$‎ </p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">تبدیل به یک جبر باناخ نمایند. همان‌طور که می‌دانیم فضای دوگان دوم جبر باناخ ‎$A$‎، یعنی ‎$A''$‎، فضایی باناخ می‌باشد. امّا برای تبدیل شدن به جبر باناخ نیازمند به تعریف ضربی بر ‎$A''$‎ می‌باشیم که شرایط لازم یک جبر را دارا باشد و همچنین نرم مفروض بر ‎$A''$‎ نیز با آن عمل ضرب، یک نرم جبری گردد‎.\\‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">ریچارد آرنز</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\LTRfootnote{Arens}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> ‎$(1919-2000)$‎ در سال ‎$1951$‎ دو ضرب بر ‎$A''$‎، که بعدها به‌عنوان ضرب آرنز</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">\index{ضرب آرنز}</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> اول و دوم معروف شدند، </p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">تعریف نمود که فضای باناخ ‎$A''$‎ با هر کدام از این ضرب‌ها به جبر باناخ تبدیل می‌شود</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎(\lr{\cite[p‎. ‎249]{6.2}}‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">و</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\lr{\cite[p‎. ‎50]{7.2}}‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">را ملاحظه کنید).</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">بنابه تعریف، ضرب آرنز</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">\index{ضرب آرنز}</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> اول ‎\scalebox{0.8}{$\square$}‎ روی</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> ‎$A^{\prime\prime}$‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">توسط ساختار ‎$A$-‎مدول چپ روی ‎$A$‎  تولید می‌شود. به‌ازای هر ‎$\Phi‎, ‎\Psi\in A^{\prime\prime} $‎  و هر</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> ‎$f\in A^{\prime}$‎ و </p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎$a,b\in A$‎ </p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">داریم:</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎$$\left&lt;\Phi \scalebox{0.8}{$\scalebox{0.8}{$\square$}$} \Psi,f\right&gt;=\left&lt;\Phi‎, ‎\Psi \cdot f\right&gt;,\quad \left&lt;\Psi \cdot f,a\right&gt;=\left&lt;\Psi,f \cdot a\right&gt;,\quad \left&lt;f\cdot a,b\right&gt;=\left&lt;f,ab\right&gt;.$$‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">به‌طور مشابه، ضرب آرنز دوم ‎$\diamondsuit$‎ روی  ‎$A^{\prime\prime}$‎ با در نظر گرفتن ‎$A$‎ به‌عنوان ‎$A$-‎مدول راست تعریف می‌شود. لذا تعریف می‌کنیم:</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎$$\left&lt;\Phi \diamondsuit \Psi,f\right&gt;=\left&lt;\Psi,f \cdot \Phi\right&gt;,\quad \left&lt;f \cdot \Phi,a\right&gt;=\left&lt;\Phi‎, ‎a\cdot f\right&gt;,\quad \left&lt;a\cdot f,b\right&gt;‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎=\left&lt;f,ba\right&gt;.$$‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">اکنون این سؤال مطرح می‌شود که چرا ‎$A''$‎ تحت اعمال ‎\scalebox{0.8}{$\square$}‎ و  ‎$\diamondsuit$ (ضرب‌های آرنز) بسته است؟‎\\‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">فرض ‌کنیم ‎$ F‎, ‎G\in (A'',\scalebox{0.8}{$\square$})$‎. نشان می‌دهیم ‎$F\scalebox{0.8}{$\square$} G\in A''$‎. بدین منظور کافی است نشان دهیم نگاشت ‎$ F\scalebox{0.8}{$\square$} G$‎ بر ‎$A'$‎ خطی و کراندار است. برای خطی بودن کافی است به‌ازای هر ‎$ f,g\in A'$‎ و ‎$\lambda \in \mathbb{C}$‎  رابطه زیر برقرار باشد:</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎$$\left&lt;F\scalebox{0.8}{$\square$} G,\lambda f+g\right&gt;= \lambda \left&lt;F\scalebox{0.8}{$\square$} G,f\right&gt;+\left&lt;F\scalebox{0.8}{$\square$} G,g\right&gt;.$$‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">بنابه تعریف ضرب که در بالا آمده است داریم:</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\begin{align*}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\left&lt;F\scalebox{0.8}{$\square$} G,\lambda f+g\right&gt;&amp;=\left&lt;F,G\cdot(\lambda f+g)\right&gt;=\left&lt;F,\lambda(G\cdot f)+G\cdot g\right&gt;\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎&amp;=\lambda \left&lt;F,G\cdot f\right&gt;+\left&lt;F,G\cdot g\right&gt;=\lambda \left&lt;F\scalebox{0.8}{$\square$} G,f\right&gt;+\left&lt;F\scalebox{0.8}{$\square$} G‎, ‎g\right&gt;‎.</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{align*}‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">برای کراندار بودن ‎$F\scalebox{0.8}{$\square$} G$‎ باید   ‎$M&gt;0$‎ یافت شود که به‌ازای هر ‎$ f\in A'$‎ داشته باشیم </p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎$\|(F\scalebox{0.8}{$\square$} G)(f)\|\leqslant M \,\|f\|$‎.  چون:</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎$$\| F(G\cdot f)\| \leqslant \| F \| \‎, ‎\| G\cdot f\|,$$‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">و از طرفی داریم:</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:2;">‎$$\| G\cdot f\|= \sup_{\|a\|\leqslant 1} |(G\cdot f)(a)|=\sup_{\|a\|\leqslant 1} |G(f\cdot a)| \leqslant \| G\|‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\sup_{\|a\|\leqslant 1} \| f \cdot a\|,$$‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">و به‌ازای هر ‎$b\in A$‎ و هر ‎$a$‎ در گوی یکه‌ی ‎$ A$‎ داریم:</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎$$|(f\cdot a)(b)|=|f(ab)| \leqslant \|f\| \‎, ‎\|ab\| \leqslant \|f\| \‎, ‎\|b\|\Rightarrow \|f\cdot a\|=\|f\|,$$‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">لذا نتیجه می‌شود ‎$\|G\cdot f\| \leqslant \|G\| \‎, ‎\|f\|$‎   و بنابراین:</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎$$\| (F\scalebox{0.8}{$\square$} G)(f)\|=\| F(G\cdot f)\| \leqslant \| F\| \‎, ‎\| G\cdot f\| \leqslant \|F\| \‎, ‎\|G\| \‎, ‎\|f\|.$$‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">با در نظر گرفتن ‎$M=\|F\| \‎, ‎\|G\|$‎ نتیجه می‌شود ‎$F\scalebox{0.8}{$\square$} G $‎ بر ‎$ A'$‎ کراندار است. پس ‎$A''$‎ تحت عمل ‎\scalebox{0.8}{$\scalebox{0.8}{$\square$}$}‎ بسته است. از طرفی</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> چون </p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> ‎$\|F\scalebox{0.8}{$\square$} G\|=\sup_{\|f\|\leqslant1} |(F\scalebox{0.8}{$\square$} G)(f)|$‎، پس </p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎$$\| F\scalebox{0.8}{$\square$} G\| \leqslant \|F\| \‎, ‎\|G\|.$$‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">بنابراین ‎$ (A'',\scalebox{0.8}{$\square$})$‎ یک جبر نرم‌دار است‎.\\‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">حال نشان می‌دهیم ‎$A''$‎ تحت عمل ضرب ‎$ \diamondsuit$‎  نیز بسته است. یعنی به‌ازای هر ‎$ F,G\in A''$‎ ، ‎$ F\diamondsuit G\in A''$‎  مشابه بالا نشان می‌دهیم نگاشت ‎$F\diamondsuit G$‎ خطی و کراندار است. بدین منظور فرض می‌کنیم ‎$ f,g \in A'$‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">و ‎$ \lambda \in \mathbb{C}$‎. دراین‌صورت</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\begin{align*}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\left&lt;F\diamondsuit G,\lambda f+g\right&gt;&amp;=\left&lt;G,(\lambda f+g) \cdot F\right&gt;=\left&lt;G,\lambda (f\cdot F)‎+ ‎g\cdot F\right&gt;‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎=\lambda \left&lt;G,f\cdot F\right&gt;+\left&lt;G,g\cdot F\right&gt;\\‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> ‎&amp;=\lambda \left&lt;F\diamondsuit G,f\right&gt;+\left&lt;F\diamondsuit G,g\right&gt;‎.</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{align*}‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">پس نگاشت ‎$F\diamondsuit G$‎  بر  ‎$A'$‎خطی است. حال نشان می‌دهیم نگاشت ‎$F\diamondsuit G$‎ بر ‎$A'$‎ کراندار است. نشان می‌دهیم عددی مثبت مانند  ‎$M$‎ یافت می‌شود به‌طوری‌که به‌ازای هر ‎$f\in A'$‎ داریم</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> ‎$ \|\left&lt;F\diamondsuit G,f\right&gt;\| \leqslant M \‎, ‎\|f\|$‎. بنابه تعریف نرم:</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎$$\|\left&lt;F\diamondsuit G,f\right&gt;\|=\|(F\diamondsuit G)(f)\|= \|G(f\cdot F)\| \leqslant \|G\| \‎, ‎\| f \cdot F \|.$$‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">از طرفی به‌ازای هر ‎$a\in A$‎ با توجه به تعریف نرم:</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\begin{align*}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\| f\cdot F\|= \sup_{\|a\| \leqslant 1}|(f\cdot F)(a)|&amp;=\sup_{\|a\| \leqslant 1} |F(a\cdot f)| \leqslant‎ </p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\| F \|\‎, ‎\sup_{\|a\| \leqslant 1} \| a\cdot f\|\\[2mm]‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎&amp;=\| F\| \,\sup_{\|a\| \leqslant 1} \sup_{\|b\| \leqslant 1} |(a\cdot f)(b)|\leqslant \|F\|\‎, ‎\|f\|‎,</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{align*}‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">زیرا داریم:</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎$$|(a\cdot f)(b)| =|f(ba)|\leqslant \|f\|\‎, ‎\|ba\| \leqslant \|f\|\‎, ‎\|b\|\‎, ‎\|a\|,$$‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">و در نتیجه:</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎$$\|(F\diamondsuit G)(f)\|\leqslant \|G\| \‎, ‎\| f \cdot F \| \leqslant \|G\|\‎, ‎\|F\|\,\|f\|.$$‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">حال اگر قرار دهیم  ‎$M=\|G\|\‎, ‎\|F\|$‎، آنگاه برای هر ‎$f\in A'$‎ داریم ‎$\|(F\diamondsuit G)(f)\|\leqslant M\‎, ‎\|f\|$‎ و لذا ‎$ F\diamondsuit G$‎ کراندار است و در نتیجه ‎$F\diamondsuit G \in A''$‎. از طرفی داریم:</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎$$\|F\diamondsuit G\| =\sup_{\|f\| \leqslant 1}|(F\diamondsuit G)(f)| \leqslant \|G\| \‎, ‎\|F\|,$$‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">پس ‎$ (A'',\diamondsuit)$‎ نیز جبر نرم‌دار است. </p>
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<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\begin{thm}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\emph{\lr{\cite[Theorem A.3.46]{5.2}}}‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">فرض کنیم ‎$X$‎ و ‎$Y$‎ جبرهای نرم‌دار باشند و ‎$ T\in \mathcal{B}(X,Y)$‎. دراین‌صورت یک نگاشت یکتا مانند ‎$T':Y'\to X'$‎ وجود دارد</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">که با ضابطه‌ی ‎$T'(f)=f \circ T$‎، یعنی به‌ازای هر ‎$f\in Y'$‎ و هر ‎$x\in X$‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎$$\left( T'(f)\right)(x)=f(Tx),$$‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">تعریف می‌شود و با توپولوژی ضعیف‌ستاره روی ‎$X'$‎ و ‎$Y'$‎ پیوسته است. به‌علاوه ‎$T'\in \mathcal{B}(Y',X')$‎.</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{thm}‎</p>
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<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\begin{proof}‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">فرض کنیم ‎$f\in Y'$‎ و ‎$ (f_\alpha)$‎ توری در ‎$Y'$‎ باشد که با توپولوژی ضعیف‌ستاره ‎$f_\alpha\to f$‎. برای پیوستگی ‎$ T'$‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">نشان می‌دهیم ‎$T'(f_{\alpha})\to T'(f)$‎.</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">چون همگرایی در توپولوژی ضعیف‌ستاره مد نظر است  لذا کافی است که به‌ازای هر ‎$x\in X$‎، ‎$(T'(f_{\alpha}))(x)\to (T'f)(x) $‎ که بنابه تعریف معادل است با اینکه</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎$f_{\alpha}(Tx)\to f(Tx)$‎</p>
<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> که بنابه فرض برقرار است.</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{proof}‎</p>
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<p dir='rtl' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">... (ادامه مطالب حذف شده است)</p></body></html>