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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22 changed files with 60 additions and 60 deletions

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@@ -27,7 +27,7 @@
The set $RS([a, b], G)$ is the vector space of all \textbf{Riemann-Stieltjes integrable functions} with respect to $G$.
\end{definition}
\begin{lemma}[Summation by Parts, {{\cite[Proposition 1.4]{Lang}}}]
\begin{lemma}[Summation by Parts]
\label{lemma:sum-by-parts}
Let $[a, b] \subset \real$, $E_1, E_2, H$ be TVSs over $F \in \RC$, $E_1 \times E_2 \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map, $f: [a, b] \to E_1$, $G: [a, b] \to E_2$, and $(P, c) \in \scp_t([a, b])$, then
\[
@@ -36,7 +36,7 @@
where $P' = \seqfz[n+1]{y_j} = [a, c_1, \cdots, c_n, b]$ and $c' = \seqf[n+1]{d_j} = [x_0, \cdots, x_n]$.
\end{lemma}
\begin{proof}
\begin{proof}[Proof {{\cite[Proposition 1.4]{Lang}}}. ]
Denote $c_0 = a$ and $c_{n+1} = b$, then
\begin{align*}
S(P, c, f, G) &= \sum_{j = 1}^n f(c_j)[G(x_j) - G(x_{j - 1})]