Q:

Sequence of $L^2$-normed functions, s.t. $L^2$-norm of derivatives diverges

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.

A:

Take $f_n(x) := \sqrt{2n+1}x^n,$ which are smooth functions with
$$\|f_n\|_{L^2([0,1])} = \int_0^1f_n^2(x) dx=(2n+1)\int_0^1x^{2n}dx=1.$$
Since $f_n^\prime(x)=n\sqrt{2n+1}x^{n-1}$, we get
$$\lVert f_n^\prime\rVert_{L_2([0,1])}^2=\int_0^1n^2(2n+1)x^{2n-2}dx=\frac{n^2(2n+1)}{2n-1} \to \infty.$$

