Theorem 27.4.6 (Symmetry of Higher Derivatives).label Let $E$ be a topological vector space over $K \in \RC$, $\sigma \subset \mathfrak{B}(E)$ be an ideal 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 $\tilde \sigma$-differentiable at $x_{0} \in U$, then $D_{\sigma}^{n}f(x_{0}) \in L_{\sigma}^{(n)}(E; F)$ is symmetric.
Proof [Proposition 4.5.14, BS17]. 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 27.4.5, $\phi \circ D_{\sigma}^{n} g(x_{0}) \in L^{n}(E_{0}; K)$ 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.$\square$
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