Lemma 7

For every function $f\in \mathscr{O}(U; A)$, the integral $\int_Uf \omega$ is an element of A depending only on $\boldsymbol{a}$ and on the germ of $f$ in a neighbourhood of Sp$^n(\boldsymbol{a})$. It satisfies the inequality

(4) $\|\int f \omega \|\leqslant (\int\|\omega \|)$ sup $\|f(z)\|$,

U U $z\in$Supp($h$)

and

for every $a\in A$.

We have seen above that the integral is defined for $f\in \mathscr{C}(U; A)$ and satisfies

$\|\int f \omega \|\leqslant \int\|f\| \|\omega \|\leqslant (\int\|\omega \|)$ sup $\|f(z)\|$.

U U U $z\in$Supp($h$)

Moreover, for every $a\in A$ and every $f\in \mathscr{C}(U; A)$, we have

(INT, VI, §2, n$^o2$, Prop. 2 applied to multiplication by $a$).

Let $(h', u'_1, . . . , u'_n)$ be a sequence adapted to $\boldsymbol{a}$ such that Supp($h'$)$\subset U$. Let $\omega '$ be the associated differential form. By Lemma 6 of I, p. 57, there exists a differential form $\psi$ on $\mathbf{C}^n$ of degree $n-1$, with coefficients in A and with support contained in Supp($h$)$\cup$ Supp($h'$)$\subset U$, such that

Let $f\in \mathscr{O}(U; A)$. Since the mapping $f$ is holomorphic, we have

$i=1$

(VAR, R2, p. 24, 8.8.9) and hence

By Stokes' formula (VAR, R2, p. 48, 11.2.3), we then have

Thus the element $\int_Uf \omega$ does not depend on the choice of the sequence $(h, u_1, . . . , u_n)$.

To conclude, let us prove that $\int_Uf \omega$ depends only on the germ of $f$ in a neighbourhood of Sp$^n(\boldsymbol{a})$. Let U and $U'$ be open neighbourhoods of Sp$^n(\boldsymbol{a})$. Let $f\in \mathscr{O}(U; A)$ and $f'\in \mathscr{O}(U'; A)$ be such that $f$ and $f'$ coincide on an open neighbourhood $U''$ of Sp$^n(\boldsymbol{a})$. There exists a mapping $h$ of $\mathbf{C}^n$ into $\mathbf{C}$, indefinitely differentiable, equal to 1 in a neighbourhood of Sp$^n(\boldsymbol{a})$, and with compact support contained in $U''$ (Lemma 1 of I, p. 52), and there exists $(u_1, . . . , u_n)$ such that the sequence $(h, u_1, . . . , u_n)$ is adapted to $\boldsymbol{a}$ (Lemma 3 of I, p. 54). Let $\omega$ be the associated differential form. Since Supp($\omega$ )$\subset$ Supp($h$)$\subset U''$ (Lemma 4, b) of I, p. 54), we have

which completes the proof.

This lemma proves that there exists a unique a-linear mapping $\Theta_{\boldsymbol{a}}$ of $\mathscr{O}$(Sp$^n(\boldsymbol{a}); A$) into A such that

(5) $\Theta_{\boldsymbol{a}}(f) =n$! $_n\int\widetilde{f \omega}$

for every open set U and every representative $\widetilde{f}\in \mathscr{O}(U; A)$ of a germ $f\in \mathscr{O}$(Sp$^n(\boldsymbol{a}); A$). The linear mapping $\Theta_{\boldsymbol{a}}$ is continuous by inequality (4) and EVT, II, p. 29, Proposition 5.