Is $\mathcal{C}^\infty([0,1])$ normable?

Asked Sep 08 '17 at 19:37

Active Sep 08 '17 at 19:37

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The space $\mathcal{C}^\infty(]0,1[)$ of smooth real functions over $]0,1[$ is not normable (because it is infinite dimensional and Montel).

Consider instead the space $\mathcal{C}^\infty([0,1])$ of smooth real functions over $[0,1]$. Here, wiki says it is not normable. Is it a typo on wikipedia ? If not, why aren't $\|\cdot\|_2$ or $\|\cdot\|_\infty$ norms on $\mathcal{C}^\infty([0,1])$ ?

asked Sep 08 '17 at 19:37

Noé AC

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Those induce a weaker topology than the standard topology. – Daniel Fischer Sep 08 '17 at 19:44
Ok thanks. And what is the standard topology on $\mathcal{C}^\infty([0,1])$ ? – Noé AC Sep 08 '17 at 19:49
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The topology induced by the family of seminorms $p_n \colon f \mapsto \lVert f^{(n)}\rVert_{\infty}$. It's the topology of uniform convergence of all derivatives. That makes it a Montel space too. – Daniel Fischer Sep 08 '17 at 19:52
Ok perfect. Thanks. – Noé AC Sep 08 '17 at 19:53

Is $\mathcal{C}^\infty([0,1])$ normable?

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