# Dirac and weak convergence

1. Dec 2, 2007

### tom_rylex

1. The problem statement, all variables and given/known data
Show that if {x_k} is any sequence of points in space $$R^n$$ with $$|{x_k}| \rightarrow \infty$$, then $$\delta(x-x_k) \rightarrow 0$$ weakly

2. Relevant equations

3. The attempt at a solution
I'm still trying to grasp the concept of weak convergence for distributions. It would appear that this function doesn't converge pointwise. The distribution on a test function is
$$\int \delta(x-x_k)\theta(x)dx = \theta(x_k)$$ Does the function converge weakly to zero because $$x_k$$ approaches infinity, and therefore would be outside of the region of support of any locally integrable test function?

2. Dec 2, 2007

### Dick

I think that's correct. But what exactly is your set of test functions?

3. Dec 2, 2007

### tom_rylex

My set of test functions meet the following criteria:
* function has a finite region of support, inside of which $$\theta(x) \neq 0$$, outside of which $$\theta(x)=0$$
* $$\theta(x)$$ has derivatives of all orders.

Last edited: Dec 2, 2007
4. Dec 2, 2007

### Dick

Then you are right. Finite support is enough.