Proposition 10.3.1. Let $E, F, G$ be normed spaces and $T: E \times F \to G$ be a bilinear map. If:
For each $x \in E$, $y \mapsto T(x, y)$ is a continuous linear map from $F$ to $G$.
For each $y \in F$, $x \mapsto T(x, y)$ is a continuous linear map from $E$ to $G$.
$E$ is a Banach space.
then $T \in L^{2}(E, F; G)$.
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, $M = \sup_{y \in B_F(0, 1)}\norm{T_y}_{L(E; G)}< \infty$. Thus for any $x \in E$ and $y \in F$, $\norm{T(x, y)}_{G} \le M\norm{x}_{E}\norm{y}_{F}$.$\square$