## Bilinear Continuous Operator

Let be three Banach spaces and consider a bilinear operator . Prove that is continuous if and only if there exists a constant such that .

It is obvious that if , then is continuous. Conversely if is continuous, the set is open and it contains . Therefore, for a small enough we have . Now let different of (it’s clear that 0. Then which means that . This implies .

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Categories: Analysis, Functional Analysis
Banach

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