Proof If {${T_{j}}$} converge to T then {${D^k T_{j}}$} converges to $D^k T$ for every n-tuple k

Question

Proof that Differentitaion is a continuous operation in $D'$ : if {${T_{j}}$ }converge to T then {${D^k T_{j}}$} converges to $D^k T$ for every n-tuple k.

My version to proof it. :

Let $\Phi \in D(\mathbb{R}^n)$

$\lim\limits_{j \rightarrow \infty}\langle{D^kT_{j}},\phi\rangle$ =$\lim\limits_{j \rightarrow \infty}(-1)^{|k|}\langle{T_{j}},D^k\phi\rangle$ =$(-1)^{|k|}\langle{T},D^k\phi\rangle$ =$\langle D^k{T},\phi\rangle$

We had in lecture the following definition: A sequence of distributions ${T_{j}}$ converges to a distribution ${T}$ if $\langle{T_{j}},\phi\rangle$ converges to $\langle{T},\phi\rangle$ for every $\Phi \in D(\mathbb{R}^n)$.

Which I use in the middle of this short proof. Is this all or did I forget something?

Answer 1

It´s fine so, because we can argument with my definition.