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## Bilinear Continuous Operator

Let $X,Y,Z$ be three Banach spaces and consider a bilinear operator $F:X \times Y \to Z$. Prove that $F$ is continuous if and only if there exists a constant $C$ such that $\|F(x,y)\|_Z \leq C \|x\|_X \|y\|_Y,\ \forall x \in X,\ \forall y \in Y$.

It is obvious that if $\|F(x,y)\|_Z \leq C \|x\|_X \|y\|_Y,\ \forall x \in X,\ \forall y \in Y$, then $F$ is continuous. Conversely if $F$ is continuous, the set $K=F^{-1}(D((0,0),1))$ is open and it contains $(0,0)$. Therefore, for a $c$ small enough we have $D((0,0),c)\subset K$. Now let $x\in X,y \in Y$ different of $0$ (it’s clear that $F(\cdot,0)=F(0,\cdot)=0$0. Then $\left(\frac{c x}{2\|x\|},\frac{cy}{2\|y\|}\right)\in K$ which means that $\frac{c^2}{4\|x\|\|y\|}F(x,y)\leq 1$. This implies $F(x,y)\leq \frac{4}{c^2} \|x\|\|y\|,\ \forall x \in X,\ y \in Y$.