Show that the map $f \mapsto \hat{f}$ is a continuous linear map from the normed vector space $(L^1(\mathbb{R}), \|\cdot\|_1)$ to the normed vector space $(L^\infty(\mathbb{R}), \|\cdot\|_\infty)$.
Show that the map $f \mapsto \hat{f}$ is a continuous linear map from the normed vector space $(L^1(\mathbb{R}), \|\cdot\|_1)$ to the normed vector space $(L^\infty(\mathbb{R}), \|\cdot\|_\infty)$.