\section{Multilinear Maps}
\label{section:normed-multilinear}

\begin{proposition}
\label{proposition:bilinear-separate}
    Let $E, F, G$ be normed vector 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 \hyperref[Uniform Boundedness Principle]{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}
