# Differential Operator Norm

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• Nov 28th 2011, 04:11 PM
mia25
Differential Operator Norm
Any Idea that help in solving this problem is Highly appreciated :)
• Nov 29th 2011, 07:13 AM
girdav
Re: Differential Operator Norm
Consider a function $\phi$ smooth, such that \int_{\mathbb R}\phi^2(t)dt=1, and put $\phi_n(t)=\sqrt n\phi(nt)$. Then $\lVert \phi_n\rVert_{L^2}=1$, but $\frac d{dt}\phi_n(t)=n^{\frac 32}\phi(nt)$ has a norm $L^2$ which cannot be bounded by a constant which doesn't depend on $n$.

For the second problem, note that $||f||^2\geq ||\frac d{dt}f||^2_{L^2}$.