Showing that $||\hat{f}||_{\infty} \leq ||f||_1$ in $L^1$

Question

Let $f \in L^1(\mathbb{R}^n)$ then

$\hat{f} \in L^{\infty}(\mathbb{R}^n)$ and $||\hat{f}||_{\infty} \leq ||f||_1$

How do you prove this or where can I find a proof of this fact?

Answer 1

This is about the first thing proved in any text on the Fourier transform: $$ |\hat f(\xi)|=\Bigl|\int_{\mathbb{R}^n}e^{ix\xi}\,f(x)\,dx\Bigr|\le\int_{\mathbb{R}^n}|f(x)|\,dx=\|f\|_1. $$