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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.


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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 $$\lVert f_n^\prime\rVert_{L_2([0,1])}^2=\int_0^1n^2(2n+1)x^{2n-2}dx=\frac{n^2(2n+1)}{2n-1} \to \infty.$$

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Thanks for this example! –  AlexE Sep 13 '11 at 11:59

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