Let $f_n(x) = e^{2\pi i n x}$. Question: Is $f_n, f'_n \in L^2([0,1])$ for every $n \in \mathbb{Z}$?
$f_n \in L^2([0,1])$ is true since $\| f_n \|^2_{L_2} =\int_0^1 |f_n|^2dx = 1. $
Is $f'_n \in L^2([0,1])$ for every $n\in\mathbb{Z}$? It is not since $\| f'_n \|^2_{L_2} =\int_0^1 |f'_n|^2dx = 4\pi^2 n^2, $ right?
This is a follow-up question from Definite integral implies integrand is the same