Adjusted citation formats. Moved citation off of named theorems if possible.
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Bokuan Li
2026-03-19 23:58:16 -04:00
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Let $x = x_{n_1} + \limv{N}\sum_{k = 1}^N (x_{n_{k+1}} - x_{n_k})$, then $x = \limv{k}x_{n_k} \in E$. Since $\seq{x_n}$ is a Cauchy sequence that admits a convergent subsequence, it is convergent.
\end{proof}
\begin{theorem}[Successive Approximations {{\cite[Section III.2]{SchaeferWolff}}}]
\begin{theorem}[Successive Approximations]
\label{theorem:successive-approximations}
Let $E, F$ be metric TVSs over $K \in \RC$ with pseudonorm $\rho$ and $\eta$, respectively. Let $T \in L(E; F)$, $r > 0$, $\gamma \in (0, 1)$, and $C \ge 0$. Suppose that for every $y \in B_F(0, r)$, there exists $x \in E$ such that:
\begin{enumerate}
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\]
\end{theorem}
\begin{proof}
\begin{proof}[Proof {{\cite[Section III.2]{SchaeferWolff}}}. ]
Let $y_0 = y$ and $x_0 = 0$. Let $N \in \natz$ and suppose inductively that $\seqf[N]{x_n} \subset E$ has been constructed such that:
\begin{enumerate}
\item[(I)] $\sum_{n = 1}^N\rho(x_n) \le C\eta(y)\sum_{n = 0}^{N-1}\gamma^{n}$.

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\section{The Dual Space}
\label{section:tvs-dual}
\begin{proposition}[Polarisation, {{\cite[Proposition 5.5]{Folland}}}]
\begin{proposition}[Polarisation]
\label{proposition:polarisation-linear}
Let $E$ be a vector space over $\complex$, $\phi \in \hom(E; \complex)$, and $u = \re{\phi}$, then
\begin{enumerate}
@@ -10,7 +10,7 @@
\end{enumerate}
Conversely, if $u \in \hom(E; \real)$ and $\phi \in \hom(E; \complex)$ is defined by $\dpb{x, \phi}{E} = \dpb{x, u}{E} - i \dpb{ix, u}{E}$ for all $x \in E$, then $f \in \hom(E; \complex)$.
\end{proposition}
\begin{proof}
\begin{proof}[Proof {{\cite[Proposition 5.5]{Folland}}}. ]
(1): Given that $\phi \in \hom(E; \real)$, $u \in \hom(E; \real)$. For any $x \in E$,
\[
\im{\dpb{x, \phi}{E}} = \re{-i \dpb{x, \phi}{E}} = -\dpb{ix, u}{E}