Does there exist a sequence of smooth functions $(f_n) \subset C^\infty([0,1])$ with $\|f_n\|_{L^2} = 1$ for all $n$ but $\|f^\prime_n\|_{L^2} \to \infty$?
I looked already at approximations of the Dirac function, but they are all stated with the property $\|g_\epsilon\|_{L^1} = 1$, i.e. normed w.r.t the $L^1$-norm, and I couldn't adopt them.
Thanks.