Let $x_{0}\in\mathbb{R}^{n}$ and consider translation by $x_{0}$
$$t_{x_{0}}:C_{0}^{\infty}(\mathbb{R}^{n})\to C_{0}^{\infty}(\mathbb{R}^{n})$$
defined by $t_{x_{0}}=\varphi(\cdot-x_{0}).$
Show that $t_{x_{0}}$ is linear and continuous, and $t_{x_{0}}$ can be extended to continous linear application $S(\mathbb{R}^{n})\to S(\mathbb{R}^{n}).$
My attempt: $t_{x_0}$ is linear its trivial. Now, let $\{\varphi_{j}\}\subset C_{0}^{\infty}(\mathbb{R}^{n})$ such that $\varphi_{j}\to \varphi\in C_{0}^{\infty}(\mathbb{R}^{n})$ in topology of this space. That is, exist $K\subset\mathbb{R}^{n}$ compact such that $\operatorname{supp}(\varphi_{j})\subset K, \ \operatorname{supp}(\varphi)\subset K;$
And for all $\displaystyle\alpha \in \mathbb{Z}_{+}^{n}: \lim_{j\to\infty}\sup_{x\in K} |\partial^{\alpha}\varphi_{j}(x)-\partial^{\alpha}\varphi(x)|=0.$
Now consider the compact $\{x_{0}\}+K$, we have $\operatorname{supp}(t_{x_{0}}(\varphi_{j}))\subset x_{0}+K$ and $\operatorname{supp}(t_{x_{0}}(\varphi))\subset x_{0}+K$
And for all $\displaystyle\alpha \in \mathbb{Z}_{+}^{n}: \lim_{j\to\infty}\sup_{x\in x_{0}+ K} |\partial^{\alpha}\varphi_{j}(x)-\partial^{\alpha}\varphi(x)|=0.$
Therefore $t_{x_{0}}(\varphi_{j})\to t_{x_{0}}(\varphi).$
It's correct? And for second part, can anybody help me?