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Dear all,

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.

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.

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