= Solution
The <Schwartz space> on the <real line> is
$$
\mathcal S(\mathbb R)
=\left\{\varphi\in C^\infty(\mathbb R):
p_{m,n}(\varphi)<\infty\text{ for every }m,n\in\mathbb N_0\right\},
\qquad
p_{m,n}(\varphi)=\sup_{x\in\mathbb R}|x^m\varphi^{(n)}(x)|.
$$
These <seminorms> define its <Fréchet space> topology. The space of <tempered distribution>[tempered distributions] is its <continuous dual space>,
$$
\mathcal S'(\mathbb R)=\mathcal S(\mathbb R)^*.
$$
With the angular-frequency convention, the <Fourier transform> of a <Schwartz function> is
$$
\widehat\varphi(\lambda)
=\int_{\mathbb R}e^{-i\lambda x}\varphi(x)\,dx.
$$
It maps $\mathcal S$ continuously to itself. The transform of $u\in\mathcal S'$ is defined through the <dual pairing>:
$$
\boxed{\langle\widehat u,\varphi\rangle
=\langle u,\widehat\varphi\rangle},
$$
up to the fixed reflection and $2\pi$ factor if the inverse-transform convention is used for the test function.
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