Proposition 11

— Let $F, G, H$ be three Banach spaces, $\Phi$ a continuous bilinear mapping of $F \times G$ into $H$. Let $m$ be a majorizable vectorial measure on $T$, with values in $G$. Then there exists one and only one continuous linear mapping $I_{\Phi,m}$ of $\overline{\mathscr{L}}^1_F(|m|)$ into $H$ such that, for every $z \in F$ and every numerical function $h$ integrable for $|m|$, one has $I_{\Phi,m}(hz) = \Phi(z, \int h dm)$. *Moreover,

for every function $f \in \overline{\mathscr{L}}^1_F(|m|)$.

If there exists such a mapping, its value for a step function $f$ over the $|m|$-integrable sets is uniquely determined: for, it is known that one can then write $f = \sum_i a_i \varphi_{X_i}$, where the $X_i$ are $|m|$-integrable and disjoint, and the $a_i \in F$ (Ch. IV, §4, No. 9, Lemma). The value of $I_{\Phi,m}(f)$ must therefore be equal to $\sum_i \Phi(a_i, \int \varphi_{X_i} dm)$. Now, we have (No. 3, Prop. 5)

which shows first of all that the element $\sum_i \Phi(a_i, \int \varphi_{X_i} dm)$ of $H$ does not depend on the particular expression of $f$ in the form $\sum_i a_i \varphi_{X_i}$, hence that we may denote it by $I_{\Phi,m}(f)$. One verifies immediately that the mapping $I_{\Phi,m}$ so defined is linear on the space $\mathscr{E}_F$ of step functions over the $|m|$-integrable sets: for, it suffices to write two functions $f, g$ of $\mathscr{E}_F$ in the form $f = \sum_i a_i \varphi_{X_i}$ and $g = \sum_i b_i \varphi_{X_i}$ with the same finite family of pairwise disjoint $|m|$-integrable sets $X_i$ (which is possible by virtue of the Lemma of Ch. IV, §4, No. 9). The inequality (5) then shows that $I_{\Phi,m}$ is continuous on $\mathscr{E}_F$, and since this subspace is dense in $\overline{\mathscr{L}}^1_F$ (Ch. IV, §4, No. 10, Cor. 1 of Prop. 19 and Ch. V, §1, No. 3), one deduces from this the existence and uniqueness of $I_{\Phi,m}$, as well as the inequality (4).

One says that $I_{\Phi,m}(f)$ is the integral of $f$ with respect to $m$ (relative to the bilinear mapping $\Phi$); when the value of the bilinear mapping $\Phi$ at the point $(x,y)$ is denoted $xy$, we shall write $\int f\,dm$ instead of $I_{\Phi,m}(f)$.

With the notations of No. 6, the integral $\int \langle f, g \rangle\,d\mu$ is none other than $I_{\Phi,m}(f)$ with $\Phi(x,x') = \langle x, x' \rangle$ and $m = g \cdot \mu$.