So, let $\Omega :=B_1(0) \subset R^3$ and $\{x_n\} \subset \Omega$. If $x_n \rightarrow x \in \partial B_1(0)$, $T:= \sum_{n \in N}\delta_{x_n}$ define a distribution on $D'(\Omega)$? How can I check it?

One idea of mine was to check if there is a function $f$ so that $T_f$ can be identified with the function (with an abuse of notation).

Using the definition I'll have:

$\langle T_f,\phi \rangle = \int_{- \infty}^{+ \infty} \sum_{n \in N}\delta_{x_n} \phi dx=\sum_{n \in N}\int_{- \infty}^{+ \infty} \delta_{x_n} \phi dx$

for $\phi \in C_c^\infty(\Omega)$

But here I don't know how to formalize the check. Any tips?

  • 1
    $\begingroup$ Remember that $\phi$ has compact support inside of $\Omega$. Therefore only a finite number of $x_n$ will be inside the support of $\phi$ and actually be used in the sum. $\endgroup$ – md2perpe Oct 19 at 16:04
  • $\begingroup$ Continuity of $T$ on $\mathscr D(\Omega)$ means, by definition, that for every compact set $K$ the restriction of $T$ to $\mathscr D(K)$ is continuous. $\endgroup$ – Jochen Oct 20 at 16:07
  • $\begingroup$ Ok, but I have difficulties to apply it to this exercise; it means that it defines a function and my procedure is correct? $\endgroup$ – Alessar Oct 24 at 11:56

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