Theorem 5.3.4 (Integration by Parts).label Let $[a,b]\suf\R$, $E,F,H$ be TVSs over $K\in \curl{\R,\com}$, $f:[a,b]\to E,G:[a,b]\to F$ and suppose there exists a continuous bilinear map $E\ti F\ni (x,y)\mto xy\in H$ then $f\in RS([a,b],G)\iff G\in RS([a,b],f)$ in which case
Proof. Suppose that $f\in RS([a,b],G)$ then for each $U\in \cali{N}_{K}(0)$ there exists $P_{0}=\curl{x_j}_{0}^{n}\in \scr{P}([a,b])$ such that $S(P,c,f,G)-\integral{a}{b}fdG\in U$ for all $(P,c)\in \scr{P}_{t}([a,b])$ with $P\geq P_{0}$. Let $Q_{0}=[x_{0},x_{1},x_{1},...,x_{n},x_{n}]$ then for any $(Q,d)\in \scr{P}_{t}([a,b])$ satisfying $Q\geq Q_{0}$ we have by Lemma 5.3.3 there exists $(Q',d')\in\scr{P}_{t}([a,b])$ such that
Moreover by construction $Q'\geq P_{0}$ so we have by first observation
as desired.$\square$