# Math Help - proper map. Distributions

1. ## proper map. Distributions

Hi, I have the follow question:

Probe that, if $A$ and $B$ are closet sets in $\mathbb{R}^n$, and the restriction of the map $(x,y)\mapsto x+y$ to $A+B$ is proper, then $A+B$ is closed.

Def: Let $A_1,\dots A_m$ be closet subsets of $\mathbb{R}^n$, We shall say the restriction of the map

$\mu:\mathbb{R}^{nm}\to\mathbb{R},\mu(x^{(1)},\dots x^{(m)})=x^{(1)}+\dots +x^{(m)}$

to $A_1\times\dots\times A_m$ is proper if, for any $\delta >0$ , there is $\delta '>0$ such that $x^{(j)}\in A_j,j=1,\dots m$ and $|x^{(1)}+x^{(m)}|\leq \delta$ imply that $|x^{(j)}|\leq\delta'$ for $j=1,\dots m$.

Thanks for you help.

PS: edited: in the first line, change: proper to closed. Tnaks InvisibleMan

2. And what does "A+B proper" means?

3. Originally Posted by InvisibleMan
And what does "A+B proper" means?
ups,
I was wrong, it should say closed instead of proper.

Thanks InvisibleMan