Proposition 5.4.2 (Limit and Integral).label Let $[a,b]\suf\R$, $E,F,H$ be locally convex spaces, $E\ti F\to H$ with $(x,y)\mto xy$ be a continuous bilinear map, and $G\in BV([a,b];F)$. For each continuous seminorm $\rho$ on $E$ and $f:[a,b]\to E$, define
Let $\ang{f_\al}_{\al\in A}\suf RS([a,b],G)$ such that
- (a)
For each continuous seminorm $\rho$ on $E$, $\braks{f_\al-f }_{s,\rho}\to 0$
- (b)
$\limit{\al\in A}\integral{a }{b }f_{\al} dG$ exists.
Proof. Let $(P=\curl{x_j}_{0}^{n},c=\curl{c_j}_{1}^{n})\in \scr{P}_{t}([a,b])$, then for any continuous seminorm $\braks{\cd}_{H }$ on $H$
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 $\e>0$, then by assumptions (a) and (b), there exists $\al\in A$ such that:
- (1)
$\braks{f-f_\al}_{E }<\frac{\e}{3 \braks{G }_{\text{var},F }}$
- (2)
$\braks{\integral{a }{b }f_\al dG-\limit{\al\in A}\integral{a }{b }f_\al dG}_{H}<\frac{\e }{3}$
- (3)
Since $f_{\al}\in RS([a,b],G)$, there exists $P_{0}\in \scr{P}([a,b])$ such that for any $P\geq P_{0}$,
- (4)
$\braks{S(P,c,f_\al,G)-\integral{a }{ b }f_\al dG }_{H }\frac{\e}{3}$
$\square$