= Solution
Using <multi-index notation>, the <Schwartz space> is
$$
\mathcal S(\mathbb R^n)
=\left\{\varphi\in C^\infty:
p_{\alpha,\beta}(\varphi)
=\sup_x|x^\alpha D^\beta\varphi(x)|<\infty
\text{ for all }\alpha,\beta\right\}.
$$
Its topology is generated by the displayed <seminorms>. The <tempered distribution> $\mathcal S'(\mathbb R^n)$ is its continuous dual, with weak convergence defined by convergence of every pairing with a Schwartz function.
For the convention
$$
\widehat\varphi(\lambda)
=\int_{\mathbb R^n}e^{-i\lambda\cdot x}\varphi(x)\,dx,
$$
differentiation under the integral and <integration by parts> give
$$
D_\lambda^\alpha\widehat\varphi
=\widehat{(-ix)^\alpha\varphi},
\qquad
\lambda^\beta\widehat\varphi
=\widehat{(-iD_x)^\beta\varphi}.
$$
These identities bound every Schwartz seminorm of $\widehat\varphi$ by finitely many seminorms of $\varphi$, proving continuity. The <Fourier inversion theorem> gives the continuous inverse
$$
\varphi(x)=\frac1{(2\pi)^n}
\int e^{i\lambda\cdot x}\widehat\varphi(\lambda)\,d\lambda,
$$
so the <Fourier transform> is a continuous isomorphism of $\mathcal S$. By duality,
$$
\langle\widehat u,\varphi\rangle
=\langle u,\widehat\varphi\rangle
$$
defines a continuous isomorphism of $\mathcal S'$.
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