Bilinear function bounded by norms of arguments Let $v_1 \in X$ and $v_2 \in Y$, $v = (v_1, v_2) \in X\times Y$, $X$ and $Y$ being normed linear spaces. Let $\phi$ be a continuous bilinear function.
Why
$$\|\phi(v_1, v_2)\| \leq K\cdot\|v_1\|\cdot\|v_2\|\quad ?$$
If the function is continuous, I can understand something of the type
$$\|\phi(v_1, v_2)\|\cdot\frac1{\|v\|} \leq K$$
But I'm having trouble getting from one expression to the other.
Thanks

Hint: What can we say about the map $x\mapsto\,\big(y\mapsto \phi(x,y)\big)$? 
