= Solution
The <space of test functions> is $\mathcal D(\mathbb R)=C_c^\infty(\mathbb R)$. A sequence $\varphi_j$ converges to $\varphi$ when its supports eventually lie in a single compact set and every derivative converges uniformly. The <distribution> space $\mathcal D'(\mathbb R)$ is its continuous linear dual; weak convergence means $\langle u_j,\varphi\rangle\to\langle u,\varphi\rangle$ for every test function. We use $\langle u',\varphi\rangle=-\langle u,\varphi'\rangle$ and $\langle au,\varphi\rangle=\langle u,a\varphi\rangle$ for <multiplication of a distribution by a smooth function>.
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