If $\varphi : \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function, the support of $\varphi$ is defined by: $$\operatorname{Supp}(\varphi) = \overline{\{x \in \mathbb{R} : \varphi(x) \neq 0\}}$$ We say that $\varphi$ has compact support if $\operatorname{Supp}(\varphi)$ is a bounded subset of $\mathbb{R}$. We denote by $\mathcal{C}_{c}(\mathbb{R})$ the set of continuous functions with compact support on $\mathbb{R}$. If $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we set $$\|\varphi\|_{\infty} = \sup_{x \in \mathbb{R}} |\varphi(x)| \text{ and } \|\varphi\|_{1} = \int_{-\infty}^{+\infty} |\varphi(t)| dt$$ Show that $\|\cdot\|_{1} : \varphi \mapsto \|\varphi\|_{1}$ is a norm on $\mathcal{C}_{c}(\mathbb{R})$. One may admit without proof that $\|\cdot\|_{\infty}$ is also a norm.
If $\varphi : \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function, the support of $\varphi$ is defined by:
$$\operatorname{Supp}(\varphi) = \overline{\{x \in \mathbb{R} : \varphi(x) \neq 0\}}$$
We say that $\varphi$ has compact support if $\operatorname{Supp}(\varphi)$ is a bounded subset of $\mathbb{R}$. We denote by $\mathcal{C}_{c}(\mathbb{R})$ the set of continuous functions with compact support on $\mathbb{R}$. If $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we set
$$\|\varphi\|_{\infty} = \sup_{x \in \mathbb{R}} |\varphi(x)| \text{ and } \|\varphi\|_{1} = \int_{-\infty}^{+\infty} |\varphi(t)| dt$$
Show that $\|\cdot\|_{1} : \varphi \mapsto \|\varphi\|_{1}$ is a norm on $\mathcal{C}_{c}(\mathbb{R})$. One may admit without proof that $\|\cdot\|_{\infty}$ is also a norm.