This question has a similar that was asked before, but it's not exactly equal. Please help me with this. Let $X$ be a banach space, $Y,Z$ a normed spaces, let $B: X \times Y \to Z$ be a bilinear map, such that it's continuous in each variable, show that $B$ is continuous.
I tried to use the Uniform Boundedness Principle. For each $y\in Y$ the function $f_y:X\to Z$ defined by $f_y(x)=B(x,y)$ is continuous by assumption. Also for each $x\in X$ the function $f_x :Y\to Z$ defined by $f_x(y)=B(x,y)$ is continuous.
Since each $f_y$ is continuous $|f_y(x)|\le K_y ||x||$ and similar for every $x$ $|f_x(y)|\le K_x ||y||$
We'll use the boundedness principle on the family $f_y$ (since are defined on a Banach space). For each $x\in X$
$|f_y(x)|\le K_y ||x||$ But this I need a bound that depends only on $x$ and not on $y$
Please help me
Consider the family
$$\mathscr{F} = \{ f_y : y \in B_Y \}.$$
By the continuity of the $f_x$, the family is pointwise bounded,
$$\sup_{y\in B_Y} \lVert f_y(x)\rVert_Z = \lVert f_x\rVert < \infty.$$
By the uniform boundedness principle, the family is uniformly bounded, i.e.
$$\sup_{y\in B_Y} \lVert f_y\rVert = K < \infty.$$
And that says
$$\sup_{x\in B_X,\, y\in B_Y} \lVert f(x,y)\rVert_Z = K < \infty,$$
hence
$$\lVert f(x,y)\rVert_Z \leqslant K\cdot \lVert x\rVert_X\cdot \lVert y\rVert_Y.$$