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I'm stuck with this problem.

Assume $\Omega$ is a non-empty subset of $\mathbb{R}^n$ and $l:\mathcal{D}(\Omega)\rightarrow\mathbb{C}$ is a linear function. Then $l$ is a distribution iff for each compact $K\subset\Omega$, there exist a $m\geq 0$ integer and a constant $C_{K,m}$ such that $$|l(\phi)|\leq \sum_{|\alpha|\leq m}\sup_{x\in\Omega}|\partial^\alpha\phi(x)|, \qquad \forall\phi\in\mathcal{D}_K(\Omega)=\{\phi\in\mathcal{D}(\Omega):supp\phi\subseteq K\}.$$

What have I done so far?

  1. I proved that the above inequality implies that $\phi$ is sequentially continuous.
  2. The book says for the other direction, you need to assume that the above inequality does not always hold. Afterward, you need to construct a sequence of $\phi_n\rightarrow 0$ in $\mathcal{D}(\Omega)$, but $l(\phi_n)=1$ for each n. At this point I am stuck, and I do not know how to construct such a sequence.
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Suppose there is a compact set $K$ for which no $(m, C_{K, m})$ exist. This means that for every $m$ the ratio $$ |l(\phi)|\big/\sum_{|\alpha|\leq m}\sup_{x\in\Omega}|\partial^\alpha\phi(x)| $$ is unbounded as $\phi$ ranges over $\mathcal D_K(\Omega)$. Use this to pick $\phi_m$ such that $l(\phi_m)=1$ and $$ \sum_{|\alpha|\leq m} \sup_{x\in\Omega}|\partial^\alpha\phi(x)| \le \frac{1}{m} $$ Now you have a sequence of test functions $(\phi_m)$ that converges to $0$ in the topology of $\mathcal D(\Omega)$, but the values $l(\phi_m)$ do not tend to $0$.

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  • $\begingroup$ Great answer! Thank! $\endgroup$
    – DonQuixote
    Apr 9 '18 at 6:20

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