Timeline for Which topology for $C^\infty(X)$ works?
Current License: CC BY-SA 4.0
10 events
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| Jul 4, 2018 at 20:54 | history | edited | Sergei Akbarov | CC BY-SA 4.0 |
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| Jul 4, 2018 at 20:41 | history | edited | Sergei Akbarov | CC BY-SA 4.0 |
simplification
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| Jul 3, 2018 at 22:36 | history | edited | Sergei Akbarov | CC BY-SA 4.0 |
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| Jul 3, 2018 at 22:30 | comment | added | Sergei Akbarov | I understand. I think there must be a textbook where all the details are explained accurately. We should ask people to give a reference. | |
| Jul 3, 2018 at 22:28 | history | edited | Sergei Akbarov | CC BY-SA 4.0 |
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| Jul 3, 2018 at 17:54 | comment | added | Sergei Akbarov | The only reference that comes to my mind is Kirillov-Gvishiani springer.com/gb/book/9780387906386 But they consider only the case when $M=U$ is an open subset in ${\mathbb R}^n$. However, from their Theorem 26 in Chapter III (where they prove that $C^\infty(U)$ which they denote by ${\mathcal E}(U)$ is a Fréchet space) it follows what you need, you should just follow the scheme that I described. | |
| Jul 3, 2018 at 15:32 | comment | added | Sergei Akbarov | As a corollary, if a subspace $P\subseteq C^\infty(M)^\star$ has zero polar in $C^\infty(M)$, then its bipolar $P^{\circ\circ}$, which is the $C^\infty(M)$-weak closure of $P$, and thus the closure of $P$ in $C^\infty(M)^\star$, coincides with $C^\infty(M)^\star$. In other words, $P$ must be dense in $C^\infty(M)^\star$. Finally, we take $P$ as the linear span of delta-functions, and we obtain what we need. | |
| Jul 3, 2018 at 15:20 | comment | added | Sergei Akbarov | Arnold, I doubt that this can be seen without proving the equivalence. In my head the explanation is the following. First, we prove that $C^\infty(M)$ is a Fréchet space for $\sigma$-compact $M$. Second, we prove that for arbitrary $M$ the space $C^\infty(M)$ is isomorphic to the direct product of a family of spaces $C^\infty(M_i)$ with $\sigma$-compact $M_i$. Together this means that $C^\infty(M)$ is always stereotype. After that we conclude that the bipolar theorem (see Schaefer IV, 1.5) holds for the pair $\Big(C^\infty(M),C^\infty(M)^\star\Big)$. | |
| Jul 3, 2018 at 14:35 | comment | added | Arnold Neumaier | I like the final definition under 3. How to see from that definition that the linear combinations of delta-functionals are dense in (C∞(X))′ without first proving equivalence to the first definition? | |
| Jul 3, 2018 at 13:59 | history | answered | Sergei Akbarov | CC BY-SA 4.0 |