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Consider the set of all compactly supported distributions $v\in\mathcal{\mathcal{E}^{\prime}}(\mathbb{R}^{n})=\left(C^{\infty}\right)^{*}$ with compact support in a fixed compact set $\Omega$ . Denote this set by $E:=\left\{ v\in\mathcal{\mathcal{E}^{\prime}}(\mathbb{R}^{n}):\mbox{supp}v\subset\Omega\right\}$.

Given any fixed $f\in C^{\infty}$ , can one then find a uniform bound $\left|v(f)\right|\leq C$ that holds for all $v\in E$ ?

Obviously, we have a bound for each individual $v$ , but can we somehow utilize that their support is in the same compact set to get a uniform bounded.

I'd be very grateful for any thoughts.

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Let $\Omega=[-1,1]$, $f$ - a test function such that $f(0)=1$. Take now $v_n = n\delta_0$. Obviously, $v_n\in E$, yet $(v_n,f)=n$, which can not be boudned.

To be more general, the space $\mathcal E'$ is a linear space, hence for a given $f$ you can not find a uniform bound on $(v,f)$ for all $v\in\mathcal E'$, because we can always multiply $v$ by a constant.

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