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Reading about the space $\mathcal{C}^\infty_0(\Omega)$ of all compactly supported functions, I've came across a claim that this space is not complete with respect to the family of seminorms

$$ \|\varphi\|_j = \max_{|\alpha|\leq j}\sup_{x\in \Omega} |\partial^\alpha \varphi(x)| \ , \forall \varphi \in \mathcal{C}^\infty_0(\Omega) \ , $$

but I'm not quite sure how to produce a counterexample for this. Anyway, because of this, we have to produce a topology that is not quite as simple as the one defined by those seminorms (namely, adapt the subspace topology for $\mathcal{C}^\infty_0(K)$ for each compact subspace $K \subset \Omega$). But then, if you take a covering by an increasing sequence of compact subsets $(K_n)$ in $\Omega$, it can be shown that the family of seminorms

$$ p_{j,n} (\varphi) = \|\varphi\|_{j,n} = \max_{|\alpha|\leq j} \sup_{x\in K_n} |\partial^\alpha \varphi(x)| \ , \forall \varphi \in \mathcal{C}^\infty_0 (\Omega) \ , n\in \mathbb{N} \ $$

induces a Fréchet space structure on $\mathcal{C}^\infty_0 (\Omega)$, so what is achieved by this family that is not by the first one?

Michh
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園田海未
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  • For completeness of the second space you only have to show that the derivatives of any Cauchy sequence converges uniformly on compact subsets. For the first space you need uniform convergence on the whole of $\Omega$. – Kavi Rama Murthy Jun 02 '20 at 00:11
  • None of these two systems of seminorms are good for defining the topology of $\mathcal{C}_0^{\infty}(\Omega)$. To understand the correct way of defining the topology, read https://math.stackexchange.com/questions/3510982/doubt-in-understanding-space-d-omega/3511753#3511753 – Abdelmalek Abdesselam Jun 02 '20 at 15:43

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As mentioned in a comment above, none of these systems of seminorms $||\cdot||_j$ or $p_{j,n}$ is good as far as defining the topology of $C_0^{\infty}(\Omega)=\mathscr{D}(\Omega)$. For an explanation of how to correctly define the topology see my answer

Doubt in understanding Space $D(\Omega)$

A useful rule of thumb to see if a system of seminorms is good or not is the following test. The seminorms typically make sense for arbitrary smooth function $\varphi$ on $\Omega$. The caveat is that these would then take values in $[0,\infty]$ instead $[0,\infty)$. One must have property that if all seminorms evaluated on $\varphi$ are finite then $\varphi$ has to be of compact support. If this property does not hold, one is pretty much guaranteed that the proposed system of seminorms is not the right one.

Abdelmalek Abdesselam
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  • I think I finally understood the source of the weirdness: the topology you get from the (semi)norms I described is not complete, but it turns out the topology you can construct via inductive limit on $\mathcal{D}(\Omega)$ is such that convergence behaves very well and is complete. However, this is not metrizable. It is really weird. – 園田海未 Jun 02 '20 at 19:26
  • Even if you think you already understood the issue, and if you are still interested in it, please read https://math.stackexchange.com/questions/3510982/doubt-in-understanding-space-d-omega/3511753#3511753 – Abdelmalek Abdesselam Jun 02 '20 at 20:40