Hello!

I have a problem. Somehow it doesn't seem all that tricky but I just can't get my head around it. Ok, here it goes: Let $\displaystyle X, Y, Z$ be Banach spaces and let $\displaystyle m:X\times Y\rightarrow Z$ be a map satisfying: for fixed $\displaystyle x$ in $\displaystyle X$ $\displaystyle m(-,y)$ is linear and continuous (and the same for a fixed $\displaystyle y$ in $\displaystyle Y$). How do I show that my map $\displaystyle m$ is continuous aswell.

First I was thinking that I should just simply show that it is bdd using the continuity in the two cases but I can't really manage.

Thanks!