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@@ -12,7 +12,7 @@
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\begin{enumerate}
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\item $E$ is a Banach space.
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\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 $x = \sum_{n = 1}^\infty x_n$.
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\end\{enumerate\}
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\end{enumerate}
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\end{lemma}
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@@ -45,7 +45,7 @@
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\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.
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\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.
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\item If $\sum_{n \in \natp}|x_n| < \infty$, then $\sum_{n = 1}^\infty x_n$ converges unconditionally.
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\end\{enumerate\}
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\end{enumerate}
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In other words, a series in $\real$ converges unconditionally if and only if its positive parts or its negative parts are finite.
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\end{theorem}
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@@ -8,7 +8,7 @@
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\item For each $x \in E$, $y \mapsto T(x, y)$ is a continuous linear map from $F$ to $G$.
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\item For each $y \in F$, $x \mapsto T(x, y)$ is a continuous linear map from $E$ to $G$.
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\item $E$ is a Banach space.
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\end\{enumerate\}
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\end{enumerate}
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then $T \in L^2(E, F; G)$.
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\end{proposition}
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@@ -38,13 +38,13 @@
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\begin{enumerate}
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\item[(a)] $\norm{x}_E \le C\norm{y}_F$.
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\item[(b)] $\norm{y - Tx}_F \le \gamma \norm{y}_F$.
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\end\{enumerate\}
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\end{enumerate}
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then for any $y \in F$, there exists $\seq{x_n} \subset E$ such that:
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\begin{enumerate}
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\item $\sum_{n \in \natp}\norm{x_n}_E \le C\norm{y}_F/(1 - \gamma)$.
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\item $\sum_{n = 1}^\infty Tx_n = y$.
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\end\{enumerate\}
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\end{enumerate}
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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 $Tx = y$.
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\end{theorem}
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@@ -75,7 +75,7 @@
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\begin{enumerate}
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\item For every $x \in E$, $\sup_{T \in \mathcal{T}}\norm{Tx}_F < \infty$.
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\item $E$ is a Banach space.
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\end\{enumerate\}
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\end{enumerate}
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then $\sup_{T \in \mathcal{T}}\norm{T}_{L(E; F)} < \infty$.
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\end{theorem}
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@@ -29,7 +29,7 @@
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\item $\bracs{B(x, r)|x \in E, r > 0}$.
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\item $\bracsn{\ol{B(x, r)}|x \in E, r > 0}$.
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\item Open sets in $E$ with respect to the weak topology.
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\end\{enumerate\}
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\end{enumerate}
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\end{proposition}
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