Cleanup

src/fa/rs/bv.tex

@@ -8,10 +8,12 @@
     \[
     V_{\rho, p}(f) = \sum_{j = 1}^n \rho(f(x_j) - f(x_{j - 1}))
     \]
+
     is the \textbf{variation} of $f$ with respect to $\rho$ and $P$. The supremum over all such partitions
     \[
     [f]_{\var, \rho} = \sup_{P \in \scp([a, b])}V_{\rho, P}(f)
     \]
+
     is the \textbf{total variation} of $f$ on $[a, b]$ with respect to $\rho$.
 
     If $E$ is a normed space, then the variation and total variation of $f$ is taken with respect to its norm.
@@ -45,12 +47,14 @@
     = \lim_{g, \fF}\sum_{j = 1}^n \rho(g(x_j) - g(x_{j - 1}))
     = \lim_{g \in \fF}V_{\rho, P}(g)
     \]
+
     By assumption (b), $[0, M_\rho]$ is in the filter generated by $V_{\rho, P}(\fF)$. Thus $V_{\rho, P}(f) \le M_\rho$. As this holds for all $P \in \scp([a, b])$, $V_{\rho, P}(f) \le M_\rho$, and $f \in BV([a, b]; E)$.
 
     (5): For each $n \in \nat^+$, let
     \[
     D_n = \bracs{x \in [a, b]|\forall \eps > 0, \exists y \in (x - \eps, x + \eps): \norm{f(x) - f(y)}_E \ge 1/n}
     \]
+
     then $D = \bigcup_{n \in \nat^+}D_n$ is the set of discontinuity points of $f$. If $D$ is uncountable, then there exists $N \in \nat^+$ such that $D_n$ is infinite.
 
     Fix $N \in \nat^+$. Let $E_1 = D_n \cap (a, b)$ and $I_1 = (a, b)$, then

src/fa/rs/index.tex

@@ -1,7 +1,7 @@
 \chapter{The Riemann-Stieltjes Integral}
 \label{chap:rs-integral}
 
-\input{./src/fa/rs/partition.tex}
-\input{./src/fa/rs/bv.tex}
-\input{./src/fa/rs/rs.tex}
-\input{./src/fa/rs/rs-bv.tex}
+\input{./partition.tex}
+\input{./bv.tex}
+\input{./rs.tex}
+\input{./rs-bv.tex}

src/fa/rs/partition.tex

@@ -7,6 +7,7 @@
     \[
     P = \seqfz{x_j} = [a = x_0 \le \cdots \le x_n = b]
     \]
+
     The collection $\scp([a, b])$ is the set of all partitions of $[a, b]$.
 \end{definition}
 
@@ -23,6 +24,7 @@
     \[
     \sigma(P) = \max_{1 \le j \le n}(x_j - x_{j - 1})
     \]
+
     is the \textbf{mesh} of $P$.
 \end{definition}
 

src/fa/rs/rs-bv.tex

@@ -9,9 +9,10 @@
     \[
     \braks{\int_a^bf dG}_H \le \sup_{x \in [a, b]}[f]_1 \cdot [g]_{\var, 2}
     \]
+
 \end{proposition}
 \begin{proof}
-    By \ref{proposition:tvs-convex-multilinear}, there exists continuous seminorms $[\cdot]_1$ on $E_1$ and $[\cdot]_2$ on $E_2$ such that $[xy]_H \le [x]_1[y]_2$ for all $(x, y) \in E_1 \times E_2$.
+    By \autoref{proposition:tvs-convex-multilinear}, there exists continuous seminorms $[\cdot]_1$ on $E_1$ and $[\cdot]_2$ on $E_2$ such that $[xy]_H \le [x]_1[y]_2$ for all $(x, y) \in E_1 \times E_2$.
 
     Let $(P = \seqfz{x_j}, c = \seqf{c_j}) \in \scp_t([a, b])$, then
     \begin{align*}
@@ -28,6 +29,7 @@
     \[
     [f]_{u, \rho} = \sup_{x \in [a, b]}\rho(f(x))
     \]
+
     Let $\net{f} \subset RS([a, b], G)$ such that:
     \begin{enumerate}
         \item[(a)] For each continuous seminorm $\rho$ on $E_1$, $[f_\alpha - f]_{u, \rho} \to 0$.
@@ -61,6 +63,7 @@
     \[
     \rho\paren{S(P, c, f, G) - \lim_{\alpha \in A}\int_a^b f_\alpha dG} < \eps
     \]
+
 \end{proof}
 
 \begin{proposition}
@@ -74,6 +77,7 @@
         \[
         \int_a^b fdG = \limv{n}S(P_n, t_n, f, G)
         \]
+
     \end{enumerate}
 \end{proposition}
 \begin{proof}
@@ -88,6 +92,7 @@
     \[
     \rho(S(P, c, f, G) - S(Q, d, f, G)) \le 2 \cdot \sup_{\begin{array}{c} x, y \in [a, b] \\ |x - y| < \max(\sigma(P), \sigma(Q)) \end{array}}[f(x) - f(y)]_1 \cdot [G]_{\var, 2}
     \]
+
     by passing through a common refinement. Since $f \in C([a, b]; E_1)$, this bound tends to $0$ as $\max(\sigma(P), \sigma(Q))$ tends to $0$, so $\angles{S(P, c, f, G)}_{(P, c) \in \scp_t([a, b])}$ is a Cauchy net.
 
     In addition, for any $\seq{(P_n, t_n)}$ as in (2), $\limv{n}S(P_n, t_n, f, G)$ exists by sequential completeness. Since $\angles{S(P, c, f, G)}_{(P, c) \in \scp_t([a, b])}$ is Cauchy, the limit $\lim_{(P, c) \in \scp_t([a, b])}S(P, c, f, G)$ exists as well and is equal to $\limv{n}S(P_n, t_n, f, G)$.

src/fa/rs/rs.tex

@@ -9,6 +9,7 @@
     \[
     S(P, c, f, G) = \sum_{j = 1}^n f(c_j)[G(x_j) - G(x_{j - 1})]
     \]
+
     is the \textbf{Riemann-Stieltjes sum} of $f$ with respect to $G$ and $(P, c)$.
 \end{definition}
 
@@ -20,6 +21,7 @@
     \[
     \int_a^b f dG = \int_a^b f(t)G(dt) = \lim_{(P, c) \in \scp_t([a, b])}S(P, c, f, G)
     \]
+
     exists. In which case, $\int_a^b fdG$ is the \textbf{Riemann-Stieltjes integral} of $G$.
 
     The set $RS([a, b], G)$ is the vector space of all \textbf{Riemann-Stieltjes integrable functions} with respect to $G$.
@@ -31,6 +33,7 @@
     \[
     S(P, c, f, G) + S(P', c', G, f) = f(b)G(b) - f(a)G(a)
     \]
+
     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}
@@ -53,16 +56,19 @@
     \[
     \int_a^b f dG + \int_a^b G df = f(b)G(b) - f(a)G(a)
     \]
+
 \end{theorem}
 \begin{proof}
     Suppose that $f \in RS([a, b], G)$. Let $U \in \cn_F(0)$, then there exits $P_0 = \seqfz{x_j} \in \scp([a, b])$ such that $S(P, c, f, G) - \int_a^b fdG \in U$ for all $(P, c) \in \scp_t([a, b])$ with $P \ge P_0$. Let
     \[
     Q_0 = [x_0, x_1, x_1, \cdots, x_n, x_n]
     \]
+
     then for any $(Q = \seqfz[m]{y_j}, d = \seqf[m]{d_j}) \in \scp_t([a, b])$ with $Q \ge Q_0$,
     \[
     f(b)G(b) - f(a)G(a) - \int_a^b fdG - S(Q, d, G, f) =
     \int_a^b fdG - S(Q', d', G, f)
     \]
-    by \ref{lemma:sum-by-parts}, where $d$ and $Q'$ contain $\seqfz{x_j}$. Thus $(Q', d') \ge P_0$, and $\int_a^b fdG - S(Q', d', G, f) \in U$.
+
+    by \autoref{lemma:sum-by-parts}, where $d$ and $Q'$ contain $\seqfz{x_j}$. Thus $(Q', d') \ge P_0$, and $\int_a^b fdG - S(Q', d', G, f) \in U$.
 \end{proof}