Fourier Transform of 1-Lebesgue Space Function is Bounded

Theorem
Let $n \in \N_{>0}$.

Let $\map {L^p} {\R^n}$ be the complex-valued Lebesgue $p$-space with respect to the Lebesgue measure for $p \in \R_{\ge 1} \cup \set \infty$.

Let $f \in \map {L^1} {\R^n}$.

Let $\map \FF f$ be the Fourier transform of $f$.

Then:
 * $\norm {\map \FF f}_{\map {L^\infty} {\R^n} } \le \norm f_{\map {L^1} {\R^n} }$

Proof
For each $\mathbf s \in \R^n$:

By, the claim follows.