# showing continuity

• Jan 13th 2012, 07:46 PM
mgarson
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 $X, Y, Z$ be Banach spaces and let $m:X\times Y\rightarrow Z$ be a map satisfying: for fixed $x$ in $X$ $m(-,y)$ is linear and continuous (and the same for a fixed $y$ in $Y$). How do I show that my map $m$ 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!
• Jan 14th 2012, 03:29 AM
girdav
Re: showing continuity
Do you know the principle of uniform boundedness?
• Jan 14th 2012, 05:21 AM
mgarson
Re: showing continuity
Yes I do. Is that the trick?
• Jan 14th 2012, 05:32 AM
girdav
Re: showing continuity
You have to show that $m$ is bounded on the unit ball. Put $S_X=\left\{x\in X, ||x||=1\right\}$. Then $\{m(x,\cdot)\}_{x\in S_X}$ is a family of elements of $Y^*$. Since the map $x\mapsto m(x,y)$ is linear and continuous for each $y$, the set $\{m(x,y)\}_{x\in S_X}$ is bounded for each fixed y. So $\{||m(x,\cdot)||\}$ is bounded and you can conclude.