Dear all,(adsbygoogle = window.adsbygoogle || []).push({});

I need a simple proof of the following:

Let [tex]u \in C(\mathbb{R}^3)[\tex] and [tex]\|u\|_{L^1(\mathbb{R}^3)} = 1[\tex]. For [tex]\lambda \geq 1[\tex], let us define the

transformation [tex]u\mapsto u_{\lambda}[\tex], where [tex] u_{\lambda}(x)={\lambda}^3 u(\lambda x)[\tex]. It is clear that

[tex]\|u_{\lambda}\|_{L^1(\mathbb{R}^3)} = \|u\|_{L^1(\mathbb{R}^3)} =1[\tex]. \\

How can I prove that

[tex]\lim_{\lambda\rightarrow\infty} u_{\lambda}(x)=\delta(x),[\tex] where [tex]\delta(x)[\tex] is the Dirac Delta function and

the limit is taken in the sense of distributions.

Thank you in advance.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Dirac Delta function as a limit

**Physics Forums | Science Articles, Homework Help, Discussion**