Proposition 5.4.3 (Equicontinuous and Integral).label Let $[a,b]\suf \R$, $E,F$ be locally convex spaces, $H$ be a sequentially complete locally convex space, and $E\ti F\to H$ with $(x,y)\mto xy$ be a continuous bilinear map. Let $f\in C([a,b];E),G\in BV([a,b];F)$ then
- (1)
$f\in RS([a,b];G)$
- (2)
If $\cali{F}\suf C([a,b];E)$ is equicontinuous and $\curl{(P_n,t_n)_{n\geq 1}}\suf \scr{P}_{t}([a,b])$ with $\s(P_{n})\to 0$ then
\begin{align*}\integral{a }{b }f dG=\limit{n\to\infty}S(P_{n},t_{n},f,G)\end{align*}uniformly for all $f\in \cali{F}$
Proof. Fix a continuous seminorm $\braks{\cd }_{H }$ then by Proposition 4.2.3, there exists continuous seminorms $\braks{\cd}_{E}$ on $E$ and $\braks{\cd}_{F }$ on $F$ such that $\braks{xy}_{H }\leq \braks{x }_{E }\braks{y }_{F }$ for all $(x,y)\in E\ti F$. Let $(P=\curl{x_j}_{0}^{n},c=\curl{c_j }_{1}^{n}),(Q=\curl{y_j }_{0}^{m},d=\curl{d_j }^{m}_{1})\in \scr{P}_{t}([a,b])$ with $Q\geq P$, then since each interval $[x_{j-1},x_{j}]$ is subdivided by some points of $Q$ also whenever $y_{k}\in [x_{j-1},x_{j}]$ we have $c_{j}\in [x_{j-1},x_{j}],d_{k}\in [y_{k-1},y_{k}]\suf [x_{j-1},x_{j}]$ so $\abs{c_j-d_k}\leq \s(P)$ hence we have
Therefore for any $(P,c),(Q,d)\in \scr{P}_{t}([a,b])$ we have
where $R$ is a common refine hence $\s(R)\leq\max \curl{\s(P),\s(Q)}$. Consider the filter on tagged partitions generated by the sets
If $\cali{S}:(P,c)\mto S(P,c,f,G)$ then $\cali{S}(\ang{\cali{U}_\de})$ is a Cauchy filter in $H$. Indeed, since $f\in C([a,b];E)=UC([a,b];E)$ by Proposition 5.3.7, we have $\braks{S(P,c,f,G)-S(Q,d,f,G)}_{H }\to 0$ as $\max \curl{\s(P),\s(Q)}\to 0$ by 5.1. In particular, for $\curl{(P_n,t_n)}$ as in (2), $\limit{n\to \infty}S(P_{n},t_{n},f,G)$ exists by sequential completeness of $H$ and is unique by 5.1 which proves (1). Now for $\cali{F}$ as in (2) we have by equicontinuity
so by 5.1
showing $\limit{(P,c)\in\scr{P}_t([a,b])}S(P,c,f,G)$ exists and is equal to $\limit{n\to \infty}S(P_{n},t_{n},f,G)$.$\square$