Fourier coefficient $ \hat{f}(n) $ and $\|f\|_1$

Question

If $ \hat{f}(n) $ is Fourier coefficient of $ f $, show that $ |\hat{f}(n)| \leq\|f\|_1$.

Answer 1

$|\hat f(n)| = \left|\int_0^1 f(x)e^{-2\pi inx}\,dx\right|\le\int_0^1|f(x)e^{-2\pi inx}|\,dx = \int_0^1|f(x)|\,dx = \|f\|_1$.

Answer 2

More generally: $$\left|\int_{a}^{b} f(x)\,dx\right|\leq \int_{a}^{b} |f(x)|\,dx.$$