# showing continuity

• Jan 13th 2012, 06:46 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 $\displaystyle X, Y, Z$ 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!
• Jan 14th 2012, 02:29 AM
girdav
Re: showing continuity
Do you know the principle of uniform boundedness?
• Jan 14th 2012, 04:21 AM
mgarson
Re: showing continuity
Yes I do. Is that the trick?
• Jan 14th 2012, 04:32 AM
girdav
Re: showing continuity
You have to show that $\displaystyle m$ is bounded on the unit ball. Put $\displaystyle S_X=\left\{x\in X, ||x||=1\right\}$. Then $\displaystyle \{m(x,\cdot)\}_{x\in S_X}$ is a family of elements of $\displaystyle Y^*$. Since the map $\displaystyle x\mapsto m(x,y)$ is linear and continuous for each $\displaystyle y$, the set $\displaystyle \{m(x,y)\}_{x\in S_X}$ is bounded for each fixed y. So $\displaystyle \{||m(x,\cdot)||\}$ is bounded and you can conclude.