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$\mathcal D(R)$ is the space of smooth compactly supported functions. Define the convergence on this space: $\varphi_n \to \varphi$ if

  • for all $k \in \mathbb{N} \cup \{0\}$ the sequence $\varphi_n^{(k)}$ uniformly converges to $\varphi^{(k)}$.
  • there exist a compact $K⊂R$ such that $supp(φ_n)⊂K$ for all $n \in \mathbb{N}$.

Let $\tau$ be the finest topology that ensures this convergence. Prove, that $\tau$ is not equal to the canonical LF topology (i.e. the finest locally convex topology that ensures the above convergence).

This field is quite new for me. Any help will be appreciated

Thank you in advance.

Jaimi
  • 690
  • see https://math.stackexchange.com/questions/3510982/doubt-in-understanding-space-d-omega/3511753#3511753 – Abdelmalek Abdesselam Nov 04 '21 at 19:30
  • Abdelmalek Abdesselam, I've read your answer about the topology of $D(\Omega)$, but I still can't come up with just a different topology (apart from the canonical LF topology and the trivial topology) on $D(\Omega)$, that sets the above-mentioned convergence, not to mention the finest one. Give me please a hint, if possible. – Jaimi Jan 19 '22 at 19:37

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