Adjusted citation formats. Moved citation off of named theorems if possible.

src/fa/lc/hahn-banach.tex

@@ -35,7 +35,7 @@
     \end{align*}
 \end{proof}
 
-\begin{theorem}[Hahn-Banach, Analytic Form {{\cite[Theorem 5.6, 5.7]{Folland}}}]
+\begin{theorem}[Hahn-Banach, Analytic Form]
 \label{theorem:hahn-banach}
     Let $E$ be a vector space over $K \in \RC$, $\rho: E \to \real$ be a sublinear functional, and $F \subsetneq E$ be a subspace, then
     \begin{enumerate}
@@ -43,7 +43,7 @@
         \item If $\rho$ is a seminorm, then for any $\phi \in \hom(F; \complex)$ with $\abs{\phi} \le \rho|_F$, there exists $\Phi \in \hom(E; \complex)$ such that $\abs{\Phi} \le \rho$ and $\Phi|_F = \phi$.
     \end{enumerate}
 \end{theorem}
-\begin{proof}
+\begin{proof}[Proof {{\cite[Theorem 5.6, 5.7]{Folland}}}. ]
     (1): Let $x_0 \in E \setminus F$, then by \autoref{lemma:hahn-banach}, there exists $\lambda \in \real$ such that if
     \[
     \phi_{x_0, \lambda}: F + \real x_0 \to \real \quad (x + tx_0) \mapsto x + \lambda t
@@ -96,21 +96,21 @@
     Finally, let $y \in A$, then $\dpb{y, \phi}{E} \le [y]_A < 1 = \dpb{x, \phi}{E}$.
 \end{proof}
 
-\begin{theorem}[Hahn-Banach, First Geometric Form {{\cite[Theorem 1.6]{Brezis}}}]
+\begin{theorem}[Hahn-Banach, First Geometric Form]
 \label{theorem:hahn-banach-geometric-1}
     Let $E$ be a TVS over $\real$, $A, B \subset E$ be non-empty convex sets, such that $A \cap B = \emptyset$ and $A$ is open, then there exists $\phi \in E^* \setminus \bracs{0}$ and $\alpha \in \real$ with $A \subset \bracs{\phi \le \alpha}$ and $B \subset \bracs{\phi \ge \alpha}$.
 \end{theorem}
-\begin{proof}
+\begin{proof}[Proof {{\cite[Theorem 1.6]{Brezis}}}. ]
     Let $C = A - B$, then $C \subset E$ is an open convex set by \autoref{lemma:convex-gymnastics}. Since $A \cap B = \emptyset$, $0 \not\in C$.
 
     By \autoref{lemma:hahn-banach-separation}, there exists $\phi \in E^*$ such that $C \subset \bracs{\phi < 0}$. In which case, for any $a \in A$ and $b \in B$, $\dpb{a - b, \phi}{E} < 0$ implies that $\dpb{a, \phi}{E} < \dpb{b, \phi}{E}$. Let $\alpha \in [\sup_{a \in A}\dpb{a, \phi}{E}, \inf_{b \in B}\dpb{b, \phi}{E}]$, then $A \subset \bracs{\phi \le \alpha}$ and $B \subset \bracs{\phi \ge \alpha}$.
 \end{proof}
 
-\begin{theorem}[Hahn-Banach, Second Geometric Form {{\cite[Theorem 1.7]{Brezis}}}]
+\begin{theorem}[Hahn-Banach, Second Geometric Form]
 \label{theorem:hahn-banach-geometric-2}
     Let $E$ be a locally convex space over $\real$, $A, B \subset E$ be non-empty convex sets such that $A \cap B = \emptyset$, $A$ is closed, and $B$ is compact, then there exists $\phi \in E^* \setminus \bracs{0}$, $\alpha \in \real$, and $r > 0$ with $A \subset \bracs{\phi \le \alpha - r}$ and $B \subset \bracs{\phi \ge \alpha + r}$.
 \end{theorem}
-\begin{proof}
+\begin{proof}[Proof {{\cite[Theorem 1.7]{Brezis}}}. ]
     Let $C = A - B$, then $C$ is closed by \autoref{proposition:tvs-set-operations} with $0 \not\in C$. Since $E$ is locally convex, there exists $U \in \cn^o(0)$ convex such that $U \cap C \ne \emptyset$.
 
     By \autoref{theorem:hahn-banach-geometric-1}, there exists $\phi \in E^* \setminus \bracs{0}$ and $\alpha \in \real$ such that $U \subset \bracs{f \le \alpha}$ and $C \subset \bracs{f \ge \alpha}$. In which case, for any $u \in U$, $a \in A$, and $b \in B$,

src/fa/lc/tensor.tex

@@ -54,7 +54,7 @@
 
 
 
-\begin{definition}[Cross Seminorm, {{\cite[III.6.3]{SchaeferWolff}}}]
+\begin{definition}[Cross Seminorm]
 \label{definition:cross-seminorm}
     Let $E, F$ be locally convex spaces over $K \in \RC$. For any convex circled sets $U \in \cn_E(0)$ and $V \in \cn_F(0)$, let $p: E \to [0, \infty)$ and $q: F \to [0, \infty)$ be their \hyperref[gauges]{definition:gauge}. For any $z \in E \otimes_\pi F$, let
     \[
@@ -75,7 +75,7 @@
     \end{enumerate}
     
 \end{definition}
-\begin{proof}
+\begin{proof}[Proof {{\cite[III.6.3]{SchaeferWolff}}}. ]
     (1): Let $\lambda \in K$, then for any $\seqf{(x_j,y_j)} \subset E \times F$, 
     \[
     |\lambda| \sum_{j = 1}^n p(x_j)q(y_j) = \sum_{j = 1}^n p(\lambda x_j)q(y_j)

src/fa/rs/rs.tex

@@ -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})]

src/fa/tvs/complete-metric.tex

@@ -15,7 +15,7 @@
     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}
@@ -33,7 +33,7 @@
     \]
 
 \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)] $\sum_{n = 1}^N\rho(x_n) \le C\eta(y)\sum_{n = 0}^{N-1}\gamma^{n}$.

src/fa/tvs/dual.tex

@@ -1,7 +1,7 @@
 \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}