Let $(X , {\|\cdot\|}_X)$ and $(Y , {\|\cdot\|}_Y)$ be two Banach spaces and let $B : X \times Y \to \mathbb{R}$ (or $\mathbb{C}$) be a map. We suppose that, given $y \in Y$, $f : x \mapsto B(x , y)$ is linear and continous, and given $x \in X$, $g : y \mapsto B(x , y)$ is linear and continous too. Show that $B$ is continous on $X \times Y$. My intuition says that it is enough to use the definition to prove it but I have made a mess in the attempt. Thank you very much.
Continuity on cartesian product of two Banach spaces
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real-analysis
functional-analysis
1 Answers
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Let $F_Y=\{B(x,\cdot):Y\to R | x\in X, \|x\|=1 \}$ be a collection of maps. For each fixed $y$, we have
$$\sup\nolimits_{\|x\|=1} \|B(x,y)\|=\sup\nolimits_{B(x,\cdot) \in F_Y} \|B(x,y)\| < \infty$$
Then by the Uniform Boundedness Principle:
$$\sup\nolimits_{\|x\|=1, \|y\|=1} \|B(x,y)\| < \infty.$$