diff --git a/src/dg/complex/derivative.tex b/src/dg/complex/derivative.tex index 16fa66f..c7bbe7d 100644 --- a/src/dg/complex/derivative.tex +++ b/src/dg/complex/derivative.tex @@ -34,13 +34,13 @@ \begin{theorem}[Cauchy] \label{theorem:cauchy-homotopy} - Let $E$ be a separated locally convex space over $\complex$, $U \subset \complex$ be open, $f \in C^1(U; E)$, and $\gamma, \mu \in C([a, b]; \complex)$ be closed, rectifiable paths. If $\gamma$ and $\mu$ are homotopic, then + Let $E$ be a complete separated locally convex space over $\complex$, $U \subset \complex$ be open, $f \in C^1(U; E)$, and $\gamma, \mu \in C([a, b]; U)$ be closed rectifiable paths. If $\gamma$ and $\mu$ are homotopic, then \[ \int_\gamma f = \int_\mu f \] \end{theorem} \begin{proof}[Proof of smooth case. ] - Let $\Gamma \in C^\infty([0, 1] \times [a, b]; \complex)$ be a smooth homotopy of loops from $\gamma$ to $\mu$, and + Let $\Gamma \in C^\infty([0, 1] \times [a, b]; U)$ be a smooth homotopy of loops from $\gamma$ to $\mu$, and \[ F: [0, 1] \to E \quad t \mapsto \int_{\Gamma (t, \cdot)}f = \int_a^b (f \circ \Gamma)(t, s) \Gamma(t, ds) \] @@ -93,7 +93,7 @@ \end{enumerate} Furthermore, by passing through a reparametrisation, assume without loss of generality that: - \begin{enumerate}[label=(\alph*)] + \begin{enumerate}[label=(\alph*),start=1] \item For each $t \in [0, \eps)$, $\Gamma(t, \cdot) = \gamma$. \item For each $t \in (1 - \eps, 1]$, $\Gamma(t, \cdot) = \mu$. \item For each $t \in [0, 1]$, $\Gamma$ is constant on $\bracs{t} \times ([a, a + \eps] \cup [b - \eps, b])$. @@ -138,5 +138,146 @@ \] \end{proof} +\begin{definition} +\label{definition:winding-number-1} + Let $U \subset \complex$, $z_0 \in U$, and $r > 0$ such that $\ol{B(z_0, r)} \subset U$, then the path + \[ + \omega_{z_0, r}: [0, 2\pi] \to U \quad \theta \mapsto a + re^{i\theta} + \] + + is the \textit{standard path of winding number $1$} at $a$ with radius $r$. +\end{definition} + + +\begin{theorem}[Cauchy's Integral Formula] +\label{theorem:cauchy-formula} + Let $E$ be a complete separated locally convex space over $\complex$, $U \subset \complex$ be open, $z_0 \in U$, $r > 0$ such that $\ol{B(z_0, r)} \subset U$, $\gamma \in C([a, b]; \complex)$ be a closed, rectifiable path homotopic to $\omega_{z_0, r}$ on $U \setminus \bracs{z_0}$, and $f \in C^1(U; E)$, then + \begin{enumerate} + \item $\int_\gamma f = 0$. + \item $f(z) = \frac{1}{2\pi i}\int_\gamma \frac{f(z)}{z - z_0}dz$. + \end{enumerate} + + More over, for any $g \in C(U; E)$ that satisfies (2) for all $z_0 \in U$, $r > 0$ with $\ol{B(z_0, r)} \subset U$, closed rectifiable curve $\gamma \in C([a, b]; \complex)$ homotopic to $\omega_{z_0, r}$ on $U \setminus \bracs{z_0}$, + \begin{enumerate}[start=2] + \item $g \in C^\infty(U; E)$, where for each $k \in \natz$, + \[ + D^kg(z_0) = \frac{k!}{2\pi i}\int_{\gamma} \frac{g(z)}{(z - z_0)^{k+1}}dz + \] + \end{enumerate} +\end{theorem} +\begin{proof} + By \autoref{theorem:cauchy-homotopy} and the \hyperref[change of variables formula]{theorem:rs-change-of-variables}, for any $g \in C^1(U \setminus \bracs{z_0}; E)$, + \[ + \int_\gamma g = \lim_{s \downto 0} \int_{\omega_{z_0, s}} g = \int_0^{2\pi} + = \lim_{s \downto 0}\frac{s}{2\pi} \int_{0}^{2\pi} g \circ \omega_{z_0, s}(\theta) e^{i\theta} d\theta + \] + + (1): Since $f \in C(U; E)$, $f$ is bounded on $\ol{B(z_0, r)}$, so for any $s \in (0, r)$, + \[ + \frac{s}{2\pi} \int_{0}^{2\pi} f \circ \omega_{z_0, s}(\theta) e^{i\theta} d\theta \in s\ol{\text{Conv}}(f(\ol{B(z_0, r)})) + \] + + As $E$ is locally convex, + \[ + \int_\gamma g = \lim_{s \downto 0} \int_{\omega_{z_0, s}} g = 0 + \] + + (2): Since $f \in C(U; E)$, + \begin{align*} + \frac{1}{2\pi i}\int_{\gamma} \frac{f(z)}{z - z_0}dz &= \lim_{s \downto 0}\frac{s}{2\pi} \int_{0}^{2\pi} \frac{f \circ \omega_{z_0, s}(\theta)}{\omega_{z_0, s}(\theta) - z_0} e^{i\theta} d\theta \\ + &= \lim_{s \downto 0}\frac{1}{2\pi}\int_0^{2\pi} f \circ \omega_{z_0, s}(\theta) d\theta = f(z_0) + \end{align*} + + (3): Suppose inductively that (3) holds for $k \in \natz$. For sufficiently small $h \in \complex$, + \[ + \frac{D^kg(z_0 + h) -D^kg(z_0)}{h} = \frac{k!}{2\pi ih} \int_\gamma \frac{g(z)}{(z - z_0-h)^{k+1}} - \frac{g(z)}{(z- z_0)^{k+1}}dz + \] + + By \autoref{proposition:difference-quotient-compact}, + \[ + \lim_{h \to 0}\frac{D^kg(z_0 + h) -D^kg(z_0)}{h} = \frac{(k+1)!}{2\pi i} \int_\gamma \frac{g(z)}{(z - z_0)^{k+2}} dz + \] + + Therefore $g \in C^{k+1}(U; E)$ with + \[ + D^{k+1}g(z_0) = \frac{(k+1)!}{2\pi i} \int_\gamma \frac{g(z)}{(z - z_0)^{k+2}} dz + \] +\end{proof} + +\begin{corollary}[Cauchy's Estimate] +\label{corollary:cauchy-estimate} + Let $E$ be a complete separated locally convex space over $\complex$, $U \subset \complex$ be open, $z_0 \in U$, $r > 0$ such that $\ol{B(z_0, r)} \subset U$, then for any $k \in \natz$ and continuous seminorm $[\cdot]_E: E \to [0, \infty)$, + \[ + [D^kf(z_0)]_E \le \frac{k!}{r^k} \sup_{z \in \ol{B(z_0, r)}}[f(z)]_E + \] +\end{corollary} +\begin{proof} + By \autoref[Cauchy's Integral Formula]{theorem:cauchy-formula} and \autoref{proposition:rs-bound}, + \begin{align*} + D^kf(z_0) &= \frac{k!}{2\pi i}\int_{\omega_{z_0, r}} \frac{f(z)}{(z - z_0)^{k+1}}dz \\ + [D^kf(z_0)]_E &\le \frac{k!}{2\pi i}\int_0^{2\pi}\frac{[f(z)]_E}{|z - z_0|^{k+1}}dz \\ + &= \frac{k!}{2\pi i}\int_0^{2\pi}\frac{[f(z)]_E}{r^{k+1}}dz \le \frac{k!}{r^k} \sup_{z \in \ol{B(z_0, r)}}[f(z)]_E + \end{align*} +\end{proof} + + +\begin{definition}[Complex Analytic] +\label{definition:complex-analytic} + Let $E$ be a complete separated locally convex space over $\complex$, $U \subset \complex$ be open, and $f \in C(U; E)$, then the following are equivalent: + \begin{enumerate} + \item (\textbf{Complex Differentiability}) $f \in C^1(U; E)$. + \item (\textbf{Cauchy-Riemann Equations}) Under the identification of $C = \real^2$, $\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \in C(U; E)$ and + \[ + \frac{\partial f}{\partial x} = i\frac{\partial f}{\partial y} + \] + \item (\textbf{Cauchy's Integral Formula}) For each $z_0 \in U$, $r > 0$ such that $\ol{B(z_0, r)} \subset U$, and closed rectifiable path $\gamma \in C([a, b]; U)$ homotopic to $\omega_{z_0, r}$ on $U \setminus \bracs{z_0}$, + \[ + f(z_0) = \frac{1}{2\pi i} \int_\gamma \frac{f(z)}{z - z_0}dz + \] + \item (\textbf{Analyticity}) For each $z_0 \in U$ and $r > 0$ such that $\ol{B(z_0, r)} \subset U$, there exists $\seq{a_n} \subset E$ such that $f$ may be expressed as a power series + \[ + f(z) = \sum_{n = 0}^\infty a_n(z - z_0)^n + \] + + with radius of convergence at least $r$. + \item (\textbf{Weak Holomorphy}) For each $\phi \in E^*$, $\phi \circ f$ satisfies the above. + \end{enumerate} + + If the above holds, then $f$ is \textbf{complex analytic}. +\end{definition} +\begin{proof} + (1) $\Leftrightarrow$ (2): \autoref{lemma:complex-analytic}. + + (1) + (2) $\Rightarrow$ (3): See \hyperref[Cauchy's Integral Formula]{theorem:cauchy-formula}. + + (3) $\Rightarrow$ (4): By \hyperref[Cauchy's Integral Formula]{theorem:cauchy-formula}, $f \in C^\infty(U; E)$ where for each $k \in \natz$, + \[ + D^kf(z_0) = \frac{k!}{2\pi i}\int_{\gamma} \frac{f(z)}{(z - z_0)^{k+1}}dz + \] + + Let + \[ + g(z) = \sum_{k = 0}^\infty \frac{1}{k!} D^kf(z_0)(z - z_0)^n + \] + + then by \hyperref[Cauchy's Estimate]{corollary:cauchy-estimate}, for any $k \in \natz$ and continuous seminorm $[\cdot]_E: E \to [0, \infty)$, + \[ + [D^kf(z_0)]_E \le \frac{k!}{r^k} \sup_{z \in \ol{B(z_0, r)}}[f(z)]_E = \frac{Ck!}{r^k} + \] + + Thus $[D^kf(z_0)/k!]_E \le C/r^k$ for all $k \in \natz$, and the radius of convergence of $g$ is at least $r$. + + Let $z \in B(z_0, r/2)$, $s = |z - z_0|$, and $n \in \natp$, then by \hyperref[Taylor's Formula]{theorem:taylor-lagrange} and \hyperref[Cauchy's Estimate]{corollary:cauchy-estimate}, + \begin{align*} + \braks{f(z) - \sum_{k = 0}^n \frac{1}{k!} D^kf(z_0)(z - z_0)^n}_E &\le s^{n+1} \cdot \sup_{z' \in \ol{B(z_0, s)}} [D^{n+1}f(z')]_E \\ + &\le \frac{Cs^{n+1}}{(r-s)^{n+1}} + \end{align*} + + which tends to $0$ as $n \to \infty$. + + (4) $\Rightarrow$ (1): By \autoref{theorem:termwise-differentiation}. + + (5) $\Rightarrow$ (3): By the equivalence of the prior points, for any $\phi \in E^*$, $\phi \circ f$ satisfies (3). By the \autoref[Hahn-Banach Theorem]{proposition:hahn-banach-utility}, $f$ also satisfies (3). +\end{proof} diff --git a/src/dg/derivative/euclid.tex b/src/dg/derivative/euclid.tex index f191def..8ee0de8 100644 --- a/src/dg/derivative/euclid.tex +++ b/src/dg/derivative/euclid.tex @@ -34,24 +34,24 @@ \begin{proposition} \label{proposition:difference-quotient-compact} - Let $E$ be a separated locally convex space, $Y$ be a Hausdorff space, and $f: (a, b) \times Y \to E$. If $f$ is differentiable in the first variable and $\frac{df}{dx} \in C((a, b) \times Y; E)$, then + Let $E$ be a separated locally convex space over $K \in \RC$, $U \subset K$ be open, $Y$ be a Hausdorff space, and $f: U \times Y \to E$. If $f$ is differentiable in the first variable and $\frac{df}{dx} \in C(U \times Y; E)$, then \[ \frac{f(x + h, y) - f(x, y)}{h} \to \frac{df}{dx}(x, y) \] - as $h \to 0$, uniformly on compacts. + as $h \to 0$, uniformly on compact sets. \end{proposition} \begin{proof} - Let $[c, d] \subset (a, b)$ and $K \subset Y$ be compact, then by the \hyperref[Mean Value Theorem]{theorem:mean-value-theorem-line}, for any $(x, y) \in [c, d] \times K$ and $h \in \real$ with $x + h$, + Let $A \subset U$ and $B \subset Y$ be compact, then by the \hyperref[Mean Value Theorem]{theorem:mean-value-theorem-line}, for any $(x, y) \in A \times B$ and $h \in \real$ with $x + h$, \begin{align*} &\frac{f(x + h, y) - f(x, y)}{h} - \frac{df}{dx}(x, y) \\ - &\in \overline{\text{Conv}}\bracs{\frac{df}{dx}(x + k, y) - \frac{df}{dx}(x, y) \bigg | k \in [-h, h]} + &\in \overline{\text{Conv}}\bracs{\frac{df}{dx}(x + k, y) - \frac{df}{dx}(x, y) \bigg | k \in B_K(0, |h|)} \end{align*} - Let $\eps > 0$ such that $[c - \eps, d + \eps] \subset (a, b)$, then since $\frac{df}{dx} \in C((a, b) \times Y; E)$, $\frac{df}{dx}|_{[c - \eps, d + \eps] \times K}$ is uniformly continuous\footnote{$K$ is a compact Hausdorff space, which comes with a \hyperref[unique uniform structure]{proposition:compact-uniform-structure}. }. Since $E$ is locally convex, + Let $\eps > 0$ such that $A + B_K(0, |\eps|) \subset U$, then since $\frac{df}{dx} \in C(U \times Y; E)$, $\frac{df}{dx}|_{(A + B_K(0, |\eps|)) \times B}$ is uniformly continuous\footnote{$K$ is a compact Hausdorff space, which comes with a \hyperref[unique uniform structure]{proposition:compact-uniform-structure}. }. Since $E$ is locally convex, \[ \frac{f(x + h, y) - f(x, y)}{h} - \frac{df}{dx}(x, y) \to 0 \] - uniformly on $[c, d] \times K$. -\end{proof} + uniformly on $A \times B$. +\end{proof} \ No newline at end of file diff --git a/src/dg/derivative/power.tex b/src/dg/derivative/power.tex index 28afe65..7b4f668 100644 --- a/src/dg/derivative/power.tex +++ b/src/dg/derivative/power.tex @@ -3,7 +3,7 @@ \begin{definition}[Power Series] \label{definition:power-series} - Let $E$ be a normed space over $K \in \RC$, $F$ be a Banach space over $K$, $\bracsn{T_n}_0^\infty$ with $T_n \in L^n(E; F)$ for each $n \in \natz$, and $a \in E$, then the \textbf{power series} of $\bracsn{T_n}_0^\infty$ about $a$ is the function + Let $E, F$ be locally convex spaces $K \in \RC$ with $F$ being complete, $\bracsn{T_n}_0^\infty$ with $T_n \in L^n(E; F)$ for each $n \in \natz$, and $a \in E$, then the \textbf{power series} of $\bracsn{T_n}_0^\infty$ about $a$ is the function \[ f(x) = \sum_{n = 0}^\infty T_n(x - a)^{(n)} \] @@ -13,25 +13,39 @@ \begin{definition}[Radius of Convergence] \label{definition:radius-of-convergence} - Let $E$ be a normed space over $K \in \RC$, $F$ be a Banach space over $K$, and $\sum_{n = 0}^\infty T_n(x - a)^{(n)}$ be a power series about $a \in E$, then $R \in [0, \infty]$ be defined by\footnote{Under the abuse that $1/\infty = 0$ and $1/0 =\infty$.} + Let $E$ be a normed space over $K \in \RC$, $F$ be a complete locally convex space over $K$, and $\sum_{n = 0}^\infty T_n(x - a)^{(n)}$ be a power series about $a \in E$, $\rho: F \to [0, \infty)$ be a continuous seminorm on $F$. For each $T \in L^n(E; F)$, let \[ - \frac{1}{R} = \limsup_{n \to \infty}\norm{T_n}_{L^n(E; F)}^{1/n} + [T]_{L^n(E; F), \rho} = \sup_{x \in B_E(0, 1)^n}\rho(Tx) + \] + + then $R_\rho \in [0, \infty]$ be defined by\footnote{Under the abuse that $1/\infty = 0$ and $1/0 =\infty$.} + \[ + \frac{1}{R_\rho} = \limsup_{n \to \infty}\norm{T_n}_{L^n(E; F)}^{1/n} \] - is the \textbf{radius of convergence of $\sum_{n = 0}^\infty T_n(x - a)^{(n)}$}. For each $0 < r < R$, the series converges uniformly and absolutely on $B_E(a, r)$. + is the \textbf{radius of convergence of $\sum_{n = 0}^\infty T_n(x - a)^{(n)}$} with respect to $\rho$, and + \begin{enumerate} + \item For each $0 < r < R$, the series converges uniformly and absolutely on $B_E(a, r)$ with respect to $\rho$. + \item Let + \[ + R = \inf\bracs{R_\rho| \rho: F \to [0, \infty) \text{ is a continuous seminorm}} + \] + + the series converges uniformly and absolutely on $B_E(a, R)$, and $R$ is the \textbf{radius of convergence of $\sum_{n = 0}^\infty T_n(x - a)^{(n)}$}. + \end{enumerate} \end{definition} \begin{proof} For all $x \in B_E(a, r)$, \[ - \sum_{n = 0}^\infty \normn{T_n(x - a)^{(n)}}_F \le \sum_{n \in \natz} \norm{T_n}_{L^n(E; F)} \norm{x - a}_E^n \le \sum_{n \in \natz} \norm{T_n}_{L^n(E; F)} r^n + \sum_{n = 0}^\infty \rho(T_n(x - a)^{(n)}) \le \sum_{n \in \natz} [T_n]_{L^n(E; F), \rho} \norm{x - a}_E^n \le \sum_{n \in \natz} r^n[T_n]_{L^n(E; F), \rho} \] For any $s \in (r, R)$, there exists $N \in \natp$ such that $\norm{T_n}_{L^n(E; F)}^{1/n} \le 1/s$ for all $n \ge N$. In which case, \[ - \sum_{n = 0}^\infty \norm{T_n}_{L^n(E; F)} r^n \le \sum_{n = 0}^N \norm{T_n}_{L^n(E; F)}r^n + \sum_{n \ge N}\frac{r^n}{s^n} < \infty + \sum_{n = 0}^\infty r^n[T_n]_{L^n(E; F), \rho} \le \sum_{n = 0}^N r^n[T_n]_{L^n(E; F), \rho} + \sum_{n \ge N}\frac{r^n}{s^n} < \infty \] - As this estimate holds uniformly on $B_E(a, r)$, the series converges uniformly and absolutely on $B_E(a, r)$. + As this estimate holds uniformly on $B_E(a, r)$, the series converges uniformly and absolutely on $B_E(a, r)$ with respect to $\rho$. \end{proof} \begin{remark} @@ -42,9 +56,9 @@ \begin{theorem}[Termwise Differentiation] \label{theorem:termwise-differentiation} - Let $E$ be a normed space over $K \in \RC$, $F$ be a Banach space over $K$, $f(x) = \sum_{n = 0}^\infty T_n(x - a)^{(n)}$ a power series about $a \in E$, and $R$ be its radius of convergence, then + Let $E$ be a normed space over $K \in \RC$, $F$ be a complete locally convex space over $K$, $f(x) = \sum_{n = 0}^\infty T_n(x - a)^{(n)}$ a power series about $a \in E$, and $R$ be its radius of convergence, then \begin{enumerate} - \item $f \in C^\infty(B(a, R); F)$. + \item $f \in C^\infty(B(a, R); F)$ is infinitely Fréchet differentiable. \item For each $x \in B(a, R)$ and $h \in E$, \[ Df(x)(h) = \sum_{n = 0}^\infty \sum_{k = 1}^{n+1}T_{n+1}(((x-a)^{(n)}, h)^{\bracs{k}}) @@ -54,14 +68,20 @@ \end{enumerate} \end{theorem} \begin{proof} - (3): For each $n \in \natz$, let + (3): Let $\rho: F \to [0, \infty)$ be a continuous seminorm. For each $n \in \natz$ and $T \in L^n(E; F)$, let + \begin{align*} + [T]_{L^n(E; F), \rho} &= \sup_{x \in B_E(0, 1)^n}\rho(Tx) \\ + [T]_{L^n(E; L(E; F)), \rho} &= \sup_{x \in B_E(0, 1)^n}[Tx]_{L(E; F), \rho} + \end{align*} + + and \[ S_n(x_1, \cdots, x_{n})(h) = \sum_{k = 1}^{n+1}T_{n+1}(((x_1, \cdots, x_n), h)^{\bracs{k}}) \] - then $\norm{S_n}_{L^n(E; L(E; F))} \le (n+1)\norm{T_{n+1}}_{L^{n+1}(E; F)}$. Since $(n+1)^{1/n}$ is convergent and $\{||T_n||_{L^n(E; F)}^{1/n}\}$ is bounded, + then $[S_n]_{L^n(E; L(E; F)), \rho} \le (n+1)[T_{n+1}]_{L^{n+1}(E; F), \rho}$. Since $(n+1)^{1/n}$ is convergent and $\{[T_{n+1}]_{L^{n+1}(E; F), \rho}\}_1^\infty$ is bounded, \[ - \limsup_{n \to \infty} \norm{S_n}_{L^n(E; L(E; F))}^{1/n} \le \limsup_{n \to \infty}(n+1)^{1/n}\norm{T_{n+1}}_{L^{n+1}(E; F)}^{1/n} = \frac{1}{R} + \limsup_{n \to \infty} [S_n]_{L^n(E; L(E; F)), \rho}^{1/n} \le \limsup_{n \to \infty}(n+1)^{1/n}[T_{n+1}]_{L^{n+1}(E; F), \rho}^{1/n} \le \frac{1}{R} \] so the radius of convergence of the proposed series is at least $R$. diff --git a/src/fa/notation.tex b/src/fa/notation.tex index 5d49fe3..0756355 100644 --- a/src/fa/notation.tex +++ b/src/fa/notation.tex @@ -43,8 +43,11 @@ $T_{f,\rho}(x)$ & Variation function of $f$ with respect to $\rho$. & \autoref{definition:variation-function} \\ $BV([a,b]; E)$ & Functions of bounded variation. & \autoref{definition:bounded-variation} \\ $S(P, c, f, G)$ & Riemann-Stieltjes sum $\sum_j f(c_j)[G(x_j)-G(x_{j-1})]$. & \autoref{definition:rs-sum} \\ + $\int_a^b f dG$, $\int_a^b f(t) G(dt)$ & Riemann-Stieljes integral of $f$ with respect to $G$. & \autoref{definition:rs-integral} \\ $RS([a,b], G)$ & Space of RS-integrable functions w.r.t.\ $G$. & \autoref{definition:rs-integral} \\ $\mathrm{Reg}([a,b], G; E)$ & Regulated functions w.r.t.\ $G$ on $[a,b]$. & \autoref{definition:regulated-function} \\ $\mu_G$ & Lebesgue-Stieltjes measure associated with $G$. & \autoref{definition:riemann-lebesgue-stieltjes} \\ + $\int_\gamma f$, $\int_\gamma f(z)dz$ & Path integral of $f$ with respect to $\gamma$. & \autoref{definition:path-integral} \\ + $PI([a, b], \gamma; E)$ & Space of path integrable functions with respect to $\gamma$. & \autoref{definition:path-integral} \end{tabular} diff --git a/src/fa/rs/rs-bv.tex b/src/fa/rs/rs-bv.tex index 546212b..0a974d7 100644 --- a/src/fa/rs/rs-bv.tex +++ b/src/fa/rs/rs-bv.tex @@ -3,17 +3,13 @@ \begin{proposition} \label{proposition:rs-bound} - Let $[a, b] \subset \real$, $E, F, H$ be locally convex spaces, $E \times F \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map, and $G: [a, b] \to F$. - - Let $[\cdot]_H$ be a continuous seminorm on $H$, then there exists continuous seminorms $[\cdot]_E$ on $E$ and $[\cdot]_F$ on $F$ such that for any $f \in RS([a, b], G)$, + Let $[a, b] \subset \real$, $E, F, H$ be locally convex spaces, $E \times F \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map, $G: [a, b] \to F$, and $f \in RS([a, b], G)$, then for any continuous seminorms $[\cdot]_E: E \to [0, \infty)$, $[\cdot]_F: F \to [0, \infty)$, and $[\cdot]_H: H \to [0, \infty)$ such that $[xy]_H \le [x]_E[y]_F$ for all $x \in E$ and $y \in F$, \[ \braks{\int_a^bf dG}_H \le \sup_{x \in [a, b]}[f]_E \cdot [g]_{\text{var}, F} \] \end{proposition} \begin{proof} - By \autoref{proposition:tvs-convex-multilinear}, there exists continuous seminorms $[\cdot]_E$ on $E$ and $[\cdot]_F$ on $F$ such that $[xy]_H \le [x]_E[y]_F$ for all $(x, y) \in E \times F$. - Let $(P = \seqfz{x_j}, c = \seqf{c_j}) \in \scp_t([a, b])$, then \begin{align*} [S(P, c, f, G)]_H &\le \sum_{j = 1}^n [f(c_j)[G(x_j) - G(x_{j - 1})]]_H \\