Proposition 4

Let $(F_i)_{1 \leq i \leq n}$ *be a finite family of Hausdorff locally convex spaces, $F = \prod_{i=1}^n F_i$ their product, $q_i$ ($1 \leq i \leq n$) a lower semi-continuous semi-norm on $F_i$, and $q$ the semi-norm on $F$ defined by $q(x_1, \ldots, x_n) = \sum_{i=1}^n q_i(x_i)$. If $m_i$ ($1 \leq i \leq n$) is a vectorial measure on $T$ with values in $F_i$ that is $q_i$*-majorizable*, then the measure $m = (m_1, \ldots, m_n)$ with values in $F$ is q-majorizable.

For, the dual $F'$ may be identified with $\prod_{i=1}^n F_i'$, in such a way that if $x = (x_i) \in F$, $z' = (z_i') \in F'$, then $\langle x, z' \rangle = \sum_{i=1}^n \langle x_i, z_i' \rangle$. If $|\langle x, z' \rangle| \leq q(x)$ for every $x \in F$, then in particular $|\langle x_i, z_i' \rangle| \leq q_i(x_i)$ for $1 \leq i \leq n$, and the converse is obvious, which shows that the set $A'_q$ is the product of the $A'_{q_i}$. Since by hypothesis $|z_i' \circ m_i| \leq q_i(m_i)$ for $z_i' \in A'_{q_i}$, it follows that

for every $z' \in A'_q$, which proves the proposition.

$^1$ Majorées.