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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 327 / 1 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-327/1/a/solution