Any Idea that help in solving this problem is Highly appreciated :)

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- Nov 28th 2011, 03:11 PMmia25Differential Operator Norm
Any Idea that help in solving this problem is Highly appreciated :)

- Nov 29th 2011, 06:13 AMgirdavRe: Differential Operator Norm
Consider a function $\displaystyle \phi$ smooth, such that \int_{\mathbb R}\phi^2(t)dt=1, and put $\displaystyle \phi_n(t)=\sqrt n\phi(nt)$. Then $\displaystyle \lVert \phi_n\rVert_{L^2}=1$, but $\displaystyle \frac d{dt}\phi_n(t)=n^{\frac 32}\phi(nt)$ has a norm $\displaystyle L^2$ which cannot be bounded by a constant which doesn't depend on $\displaystyle n$.

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