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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)=00. 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.

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