Today, let’s talk about the concept of convergence in the sense of distribution. We consider via the following simple example: Assume . For each , let be the function defined by
Show that in the sense of distribution as .
At first, is assumed to be continuous, therefore cannot exist in the classical sense. The point is is considered in the sense of distribution theory, that means, is a function such that the following holds
for every test function .
Recall that the statement in the sense of distribution as is equivalent to show that
Thus, what we need to prove is the following
Since is continuous and has compact support, it is reasonable to write
Furthermore, we can take the limit as
This completes the proof.