Suppose $\mathscr{S}(\mathbb{R^n})$ is the space of Schwartz functions, in which the seminorms have the form $$\left \| \varphi \right \|_{m}=\underset{\underset{x\in \mathbb{R}^{n}}{|\alpha|\leq m }}{\sup} \left ( 1+|x|^{2} \right )^{\frac{m}{2}}\left |D^{\alpha}\varphi(x) \right | \ , \ \ ( \ m=0,1,2,\cdots) . $$ Recall that a distribution $f\in \mathscr{S}'\left (\mathbb{R}^{n} \right )$ if and only if there exists an $m\in \mathbb{N}\cup \left \{ 0 \right \}$ and $C_{m}>0$ such that $$|\left \langle f,\varphi \right \rangle |\leq C_{m}\underset{\underset{x\in \mathbb{R}^{n}}{|\alpha|\leq m}}{\sup}\left ( 1+|x|^{2} \right )^{\frac{m}{2}}\left | D^{\alpha}\varphi(x) \right | \ , \ ( \ \forall \varphi\in\mathscr{S}(\mathbb{R^n})). $$ Question:
Prove that $f\in \mathscr{S'}(\mathbb{R^n})$ if and only if there exists an $m\in \mathbb{N}\cup \left \{ 0 \right \}$ and $u_{\alpha}\in L^2\left (\mathbb{R}^{n} \right ), \ \left|\alpha \right |\leq m $, such that $$ \left \langle f,\varphi\right\rangle=\sum_{\left | \alpha \right |\leq m }\int_{\mathbb{R}^{n}}u_{\alpha}(x)D^{\alpha}\varphi(x)\left ( 1+\left | x \right |^{2} \right )^{\frac{m}{2}}dx \ , \ ( \ \forall \varphi\in\mathscr{S}(\mathbb{R^n})). $$
This result can be found in a Chinses book called "Lecture Notes on Functional Analysis I". However, to me, the proof seems too brief to understand entirely.
Can anyone help me to prove it in detail ? Or any reference ? Thanks in advance.
Supplement:
$1.$ In the proof of the equivalent seminorms between $\left \| \varphi \right \|_{m}$ and $\left \| \varphi \right \|'_{m}$, I don't know why these following steps are valid:
Suppose $e=(1, 1,\cdots , 1)$ then \begin{align*} \underset{\underset{x\in \mathbb{R}^{n}}{|\alpha|\leq m }}{\sup} \left ( 1+|x|^{2} \right )^{\frac{m}{2}}\left |D^{\alpha}\varphi(x) \right | &\leq \underset{|\alpha|\leq m}{\sup}\int_{\mathbb{R}^n}\left | D^e(1+|x|^2)^{\frac{m}{2}}D^\alpha \varphi(x) \right |dx \\ &\leq C \underset{|\alpha|\leq m}{\sup}\int_{\mathbb{R}^n}(1+|x|^2)^{\frac{m}{2}}\left | D^{\alpha+e}\varphi(x) \right |dx. \end{align*} $2.$ For the necessary part of this proposition, I would like to account for what I have known and not known:
Since $f\in \mathscr{S}'\left (\mathbb{R}^{n} \right )$, we know that there exists an $m\in \mathbb{N}\cup \left \{ 0 \right \}$ and $C_m>0$ such that $$ \left | \left \langle f,\varphi \right \rangle \right |\leq C_m \left \| \varphi \right \|'_{m}, \ ( \ \forall \varphi\in\mathscr{S}\left ( \mathbb{R}^n \right )). $$ Followed by that book, then $f$ can be continuously extended onto the Banach space $\mathscr{S}_m\left ( \mathbb{R}^n \right )$ with the norm $\left \| \cdot \right \|'_m$. I know this seminorm $\left \| \cdot \right \|'_m$ forms a norm in $\mathscr{S}\left ( \mathbb{R}^n \right ).$ And I think, maybe the norm is not complete in $\mathscr{S}\left ( \mathbb{R}^n \right ).$ However, I am not sure whether the element $\varphi$ in $\mathscr{S}_m\left ( \mathbb{R}^n \right )/\mathscr{S}\left ( \mathbb{R}^n \right )$ still satisfies the following equation which has been satisfied in $\mathscr{S}\left ( \mathbb{R}^n \right )$ : $$\left \| \varphi \right \|'_m =\left (\sum_{|\alpha| \leq m}\int_{\mathbb{R}^n}(1+|x|^2)^m|D^\alpha \varphi(x)|^2dx \right )^{\frac{1}{2}}. $$
Add a few updates:
I have found a method to solve the second question. The above equation is valid for arbitrary $\varphi\in \mathscr{S}_m\left ( \mathbb{R}^n \right )$. Now, only the first one remains to be unsolved.
Edit $1$:
The first question may have been solved by me (not very strictly). The derivative $D^e(1+|x|^2)^{\frac{m}{2}}D^\alpha \varphi(x)$ should be the form $D^e\left ((1+|x|^2)^{\frac{m}{2}}D^\alpha \varphi(x) \right )$.
Edit $2$:
Even if we take the derivative of the form $D^e\left ((1+|x|^2)^{\frac{m}{2}}D^\alpha \varphi(x) \right )$ into account, I have found that this proof would become more difficult if we treat it more strictly. For the first question, can any one help me?