Polished A-A and added new lines for broken enumerates.
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@@ -38,12 +38,14 @@
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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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\begin{proof}
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@@ -73,7 +75,8 @@
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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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\begin{proof}
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