showing continuity

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 be Banach spaces and let be a map satisfying: for fixed in is linear and continuous (and the same for a fixed in ). How do I show that my map 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!

Do you know the principle of uniform boundedness?

Yes I do. Is that the trick?

You have to show that is bounded on the unit ball. Put . Then is a family of elements of . Since the map is linear and continuous for each , the set is bounded for each fixed y. So is bounded and you can conclude.