Timeline for Gelfand-Pettis integral: what does it mean for a topological vector space to "admit a dual space?"
Current License: CC BY-SA 4.0
8 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Oct 21, 2018 at 16:57 | comment | added | Andrew | math.stackexchange.com/questions/1922567/… | |
| Oct 14, 2018 at 17:54 | comment | added | Gerald Edgar | Yes, the dual of a topological vector space is (by convention) the set of all continuous linear functionals. That is, when writing "Hom", you mean Hom in the category of topological vector spaces. | |
| Oct 14, 2018 at 16:25 | comment | added | Todd Trimble | In addition, "separates points" means that the original space embeds canonically into its double dual. | |
| Oct 14, 2018 at 16:17 | comment | added | D_S | I see, "dual space" for me means $\operatorname{Hom}_{\mathbb C}(V,\mathbb C)$, but I should take it to be the continuous functions. Thank you. | |
| Oct 14, 2018 at 16:11 | comment | added | user114263 | Every topological space admits a dual space (this is just the space of continuous, linear functionals). The full statement is "assume [...] admits a dual space which separates points." Not all topological vector spaces admit a dual space which separates points. For example, $L_\frac{1}{2}[0,1]$ has only two open, convex subsets (the empty set and the whole space). Thus there is a surprising lack of continuous, linear functionals on this space. Not enough to separate points. | |
| Oct 14, 2018 at 16:05 | comment | added | j.c. | The key phrase seems to be "that separates points". | |
| Oct 14, 2018 at 16:01 | history | asked | D_S | CC BY-SA 4.0 |