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Math Help - proper map. Distributions

  1. #1
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    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
    Last edited by yemino; October 2nd 2009 at 04:59 AM.
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  2. #2
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    And what does "A+B proper" means?
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  3. #3
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    Quote Originally Posted by InvisibleMan View Post
    And what does "A+B proper" means?
    ups,
    I was wrong, it should say closed instead of proper.


    Thanks InvisibleMan
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