Let $X$ be a Banach space and let $Y$ and $Z$ be normed vector spaces (over $\mathbb{R}$ or $\mathbb{C}$). Let $B:X\times Y \to Z$ be a bilinear map. Then the following are equivalent. Then if $B$ is bounded, i.e. if there is a constant $c\ge 0$ such that $$ ||B(x,y)||_Z \le c ||x|| \ ||y||_Y, $$ for all $x\in X$ and all $x\in Y$, this means that $B$ will be continuous.
The proof of this uses the fact that the bounded on $B$ implies that $B$ is locally Lipschitz continuous, and hence continuous.
But how is $B$ locally Lipschitz continuous and how does being bounded imply this fact? The notes I am using do not give a definition of locally Lipschitz continuous so I am wondering does someone know an appropriate definition, and how the map $B$ satisfies the definition?
Furthermore, why does the bounded only imply local Lipschitz continuity instead of global Lipschitz continuity?