Timeline for Test functions with "wrong" topology not locally convex?
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13 events
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| Mar 27, 2023 at 10:48 | comment | added | Pietro Majer | @MatthiasLudewig The TVS limit of a sequence of LCTVS is still LC, in other words, it coincides with the limit made in the LCTVS category. | |
| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Jul 9, 2014 at 16:10 | vote | accept | Matthias Ludewig | ||
| Jul 4, 2014 at 18:36 | history | edited | Peter Michor | CC BY-SA 3.0 |
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| Jul 1, 2014 at 8:41 | history | edited | Peter Michor | CC BY-SA 3.0 |
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| Jun 24, 2014 at 15:37 | comment | added | Matthias Ludewig | The Remark after proposition 4.26 that you mention states that the $c^\infty$-topology is not a vector space topology on $C^\infty_c(\mathbb{R})$. I don't see however, why the $c^\infty$-topology should coincide with the inductive topology mentioned above, as you claim. I also did not find this statement anywhere in your book. In fact, I believe that this cannot be true as the inductive topology is a vector space topology, as I prove in the Edit above. Did I make a mistake in my proof? | |
| Jun 9, 2014 at 11:50 | vote | accept | Matthias Ludewig | ||
| Jun 24, 2014 at 15:33 | |||||
| Jun 7, 2014 at 18:17 | history | edited | Peter Michor | CC BY-SA 3.0 |
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| Apr 25, 2014 at 6:06 | comment | added | Matthias Ludewig | I DO know that in general. Still, this does not mean that the two topologies cannot coincide in this special case. I want a specific argument why they don't in this one very important special case! | |
| Apr 24, 2014 at 22:45 | comment | added | paul garrett | @Kofi, one point is that the colimit in the category of all TVS's is different from the colimit in the category of locally_convex TVSs. The loc-convex colimit is ("of course") locally convex. Some colimits don't exist at all in the larger category: uncountable colimits of lines... as in math.umn.edu/~garrett/m/fun/uncountable_coproducts.pdf | |
| Apr 22, 2014 at 14:37 | comment | added | Matthias Ludewig | Did I understand something wrong? By Prop. 4.26(ii), the $c^\infty$ topology is not a vector space topology on the strict inductive limit of Fréchet spaces. But the inductive topology above is a vector space topology! | |
| Apr 22, 2014 at 13:45 | comment | added | Matthias Ludewig | Thank you for your answer! But do you have an argument, why the first topology is not locally convex? | |
| Apr 22, 2014 at 12:14 | history | answered | Peter Michor | CC BY-SA 3.0 |