Theorem 18.4.3. Let $E$ be a topological vector space over $K \in \RC$, $\sigma \subset B(E)$ be an upward-directed system that includes all bounded sets contained in finite-dimensional spaces, $F$ be a separated locally convex space over $K$, $U \subset E$ be open, and $f: E \to F$ be $n$-fold $\sigma$-differentiable at $x_{0} \in U$, then $D_{\sigma}^{n}f(x_{0}) \in B_{\sigma}^{n}(E; F)$ is symmetric.
Proof. Let $\seqf{h_j}\subset E$, $E_{0}$ be the subspace generated by $\seqf{h_j}$, and $g = f|_{E_0 \cap U}: E_{0} \cap U \to F$. Since $\sigma$ includes all bounded sets contained in finite-dimensional spaces, for any $\phi \in F^{*}$, the mapping $\phi \circ g: E_{0} \cap U \to K$ is $n$-times Fréchet-differentiable, with
by the chain rule. By Theorem 18.4.2, $\phi \circ D_{\sigma}^{n} g(x_{0}) \in L^{n}(E_{0}; K)$ is symmetric. As this holds for any $\seqf{h_j}\subset E$ and $\phi \in F^{*}$, $D_{\sigma}^{n} g(x_{0}) \in B_{\sigma}^{n}(E; F)$ is symmetric by the Hahn-Banach theorem.$\square$