Symmetry of the derivative (normed/frechet).

src/fa/norm/multilinear.tex

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+\section{Multilinear Maps}
+\label{section:normed-multilinear}
+
+\begin{proposition}
+\label{proposition:bilinear-separate}
+    Let $E, F, G$ be normed spaces and $T: E \times F \to G$ be a bilinear map. If:
+    \begin{enumerate}
+        \item For each $x \in E$, $y \mapsto T(x, y)$ is a continuous linear map from $F$ to $G$.
+        \item For each $y \in F$, $x \mapsto T(x, y)$ is a continuous linear map from $E$ to $G$.
+        \item $E$ is a Banach space.
+    \end{enumerate}
+    then $T \in L^2(E, F; G)$.
+\end{proposition}
+\begin{proof}
+    For each $y \in F$, let $T_y \in L(E; G)$ be defined by $x \mapsto T(x, y)$. Let $x \in X$, then
+    \[
+    \sup_{y \in B_F(0, 1)}\norm{T_yx}_G = \sup_{y \in B_F(0, 1)}\norm{T(x, y)}_G < \infty
+    \]
+    by continuity of $y \mapsto T(x, y)$. By the Uniform Boundedness Principle (\ref{theorem:uniform-boundedness}), $M = \sup_{y \in B_F(0, 1)}\norm{T_y}_{L(E; G)} < \infty$. Thus for any $x \in E$ and $y \in F$, $\norm{T(x, y)}_G \le M\norm{x}_E\norm{y}_F$.
+\end{proof}