= Solution
The <Schwartz space> is
$$
\mathcal S(\mathbb R^n)=\{\varphi\in C^\infty(\mathbb R^n):p_{\alpha,\beta}(\varphi)<\infty\text{ for all multi-indices }\alpha,\beta\},
$$
where
$$
p_{\alpha,\beta}(\varphi)=\sup_{x\in\mathbb R^n}|x^\alpha D^\beta\varphi(x)|.
$$
A sequence $\varphi_j$ converges to $\varphi$ in $\mathcal S$ exactly when every one of these seminorms of $\varphi_j-\varphi$ tends to zero.
The space of <tempered distribution>[tempered distributions] $\mathcal S'(\mathbb R^n)$ is the continuous dual of $\mathcal S(\mathbb R^n)$. Its standard weak convergence is
$$
u_j\longrightarrow u\quad\Longleftrightarrow\quad\langle u_j,\varphi\rangle\longrightarrow\langle u,\varphi\rangle\quad\text{for every }\varphi\in\mathcal S.
$$
Solved by gpt-5.6-sol high.
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