Proposition 26.6.2.label Let $E_{1}, E_{2}$ be TVSs over $K \in \RC$, $\sigma_{1} \subset \mathfrak{B}(E_{1})$ and $\sigma_{2} \subset \mathfrak{B}(E_{2})$ be covering ideals, $F$ be a separated locally convex space over $K$, $U \subset E_{1} \times E_{2}$ be open, $f: U \to F$, and $p \ge 1$, then the following are equivalent:
- (1)
$f \in \tilde C_{\sigma_1 \otimes \sigma_2}^{p}(U; F)$.
- (2)
$D_{1} f \in \tilde C_{\sigma_1 \otimes \sigma_2}^{p-1}(U; B_{\sigma_1}(E; F))$ and $D_{2} f \in \tilde C_{\sigma_1 \otimes \sigma_2}^{p-1}(U; B_{\sigma_2}(E; F))$
If the above holds, then for any $x \in U$ and $(h_{1}, h_{2}) \in E_{1} \times E_{2}$,
Proof. (2) $\Rightarrow$ (1): For each $(x, y) \in U$ and $(h_{1}, h_{2}) \in E_{1} \times E_{2}$,
where $r_{1} \in \mathcal{R}_{\sigma_1}(E_{1}; F)$. On the other hand, by the Mean Value Theorem,
Since $D_{2}f$ is continuous and $F$ is locally convex,
where $r_{2} \in \mathcal{R}_{\sigma_1 \otimes \sigma_2}(E_{1} \times E_{2}; F)$. Therefore
$\square$