26.6 Partial Derivatives

Definition 26.6.1 (Partial Derivative).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 TVS over $K$, $U \subset E_{1} \times E_{2}$ be open, and $f: U \to F$. For each $(x_{0}, y_{0}) \in E$, let $f_{x_0}(y) = f(x_{0}, y)$ and $f_{y_0}(x) = f(x, y_{0})$ be the partial maps of $f$. If $f_{x_0}$ is $\tilde \sigma_{1}$-differentiable for each $x_{0}$, and $f_{y_0}$ is $\tilde \sigma_{2}$-differentiable for each $y_{0}$, then

and

are the partial derivatives of $f$.

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. (1)

   $f \in \tilde C_{\sigma_1 \otimes \sigma_2}^{p}(U; F)$.

2. (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 $(h_{1}, h_{2}) \in E_{1} \times E_{2}$,