Polished A-A and added new lines for broken enumerates.
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Bokuan Li
2026-05-05 01:50:35 -04:00
parent 47a7e1de68
commit 0f2e69d1f9
81 changed files with 441 additions and 185 deletions

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\begin{enumerate}
\item $E$ is a Banach space.
\item For any absolutely convergent series $\sum_{n = 1}^\infty x_n$ with $\seq{x_n} \subset E$, there exists $x \in E$ such that $x = \sum_{n = 1}^\infty x_n$.
\end{enumerate}
\end\{enumerate\}
\end{lemma}
\begin{proof}
@@ -44,7 +45,8 @@
\item If $\sum_{n \in P}x_n = \infty$ but $\sum_{n \in N}x_n > -\infty$, then $\sum_{n = 1}^\infty x_n$ converges to $\infty$ unconditionally.
\item If $\sum_{n \in N}x_n = -\infty$ but $\sum_{n \in P}x_n < \infty$, then $\sum_{n = 1}^\infty x_n$ converges to $-\infty$ unconditionally.
\item If $\sum_{n \in \natp}|x_n| < \infty$, then $\sum_{n = 1}^\infty x_n$ converges unconditionally.
\end{enumerate}
\end\{enumerate\}
In other words, a series in $\real$ converges unconditionally if and only if its positive parts or its negative parts are finite.
\end{theorem}
\begin{proof}

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@@ -8,7 +8,8 @@
\item For each $x \in E$, $y \mapsto T(x, y)$ is a continuous linear map from $F$ to $G$.
\item For each $y \in F$, $x \mapsto T(x, y)$ is a continuous linear map from $E$ to $G$.
\item $E$ is a Banach space.
\end{enumerate}
\end\{enumerate\}
then $T \in L^2(E, F; G)$.
\end{proposition}
\begin{proof}

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@@ -38,12 +38,14 @@
\begin{enumerate}
\item[(a)] $\norm{x}_E \le C\norm{y}_F$.
\item[(b)] $\norm{y - Tx}_F \le \gamma \norm{y}_F$.
\end{enumerate}
\end\{enumerate\}
then for any $y \in F$, there exists $\seq{x_n} \subset E$ such that:
\begin{enumerate}
\item $\sum_{n \in \natp}\norm{x_n}_E \le C\norm{y}_F/(1 - \gamma)$.
\item $\sum_{n = 1}^\infty Tx_n = y$.
\end{enumerate}
\end\{enumerate\}
In particular, if $E$ is a Banach space, then for every $y \in F$, there exists $x \in E$ such that $\norm{x}_E \le C\norm{y}_F/(1 - \gamma)$ and $Tx = y$.
\end{theorem}
\begin{proof}
@@ -73,7 +75,8 @@
\begin{enumerate}
\item For every $x \in E$, $\sup_{T \in \mathcal{T}}\norm{Tx}_F < \infty$.
\item $E$ is a Banach space.
\end{enumerate}
\end\{enumerate\}
then $\sup_{T \in \mathcal{T}}\norm{T}_{L(E; F)} < \infty$.
\end{theorem}
\begin{proof}

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@@ -29,7 +29,8 @@
\item $\bracs{B(x, r)|x \in E, r > 0}$.
\item $\bracsn{\ol{B(x, r)}|x \in E, r > 0}$.
\item Open sets in $E$ with respect to the weak topology.
\end{enumerate}
\end\{enumerate\}
\end{proposition}
\begin{proof}