Polished A-A and added new lines for broken enumerates.
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Compile Project / Compile (push) Failing after 12s

This commit is contained in:
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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@@ -35,7 +35,8 @@
\begin{enumerate}
\item[(a)] For each continuous seminorm $\rho$ on $E$, $[f_\alpha - f]_{u, \rho} \to 0$.
\item[(b)] $\lim_{\alpha \in A}\int_a^b f_\alpha dG$ exists.
\end{enumerate}
\end\{enumerate\}
then $f \in RS([a, b], G)$ and $\int_a^b f dG = \lim_{\alpha \in A}\int_a^b f_\alpha dG$. In particular,
\begin{enumerate}
\item If $H$ is complete, then condition (b) may be omitted.
@@ -55,11 +56,13 @@
\begin{enumerate}
\item $[f - f_\alpha]_E < \eps/(3[G]_{\text{var}, F})$.
\item $\rho\paren{\int_a^b f_\alpha dG - \lim_{\alpha \in A}\int_a^b f_\alpha dG} < \eps/3$.
\end{enumerate}
\end\{enumerate\}
Since $f_\alpha \in RS([a, b], G)$, there exists $P_0 \in \scp([a, b])$ such that if $P \ge P_0$,
\begin{enumerate}
\item[(3)] $\rho\paren{S(P, c, f_\alpha, G) - \int_a^b f_\alpha dG} < \eps/3$.
\end{enumerate}
\end\{enumerate\}
Thus for any $(P, c) \in \scp_t([a, b])$ with $P \ge P_0$,
\[
\rho\paren{S(P, c, f, G) - \lim_{\alpha \in A}\int_a^b f_\alpha dG} < \eps