Timeline for Intrinsic description of the image of $V \to V^{**}$
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Nov 7, 2011 at 20:34 | comment | added | B. Bischof | I like this question a lot. | |
| Nov 7, 2011 at 20:20 | answer | added | André Henriques | timeline score: 9 | |
| Nov 7, 2011 at 19:57 | history | edited | Martin Brandenburg | CC BY-SA 3.0 |
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| Nov 7, 2011 at 19:01 | comment | added | Moosbrugger | Yes, it carries the discrete topology. | |
| Nov 7, 2011 at 18:49 | comment | added | Martin Brandenburg | @Gerald: Sure, the maps $hom(W,V) \to \hom(V^*,W^*) \to \hom(V^**,W^**)$ are injective, but not surjective in general. By intrinsic I mean a condition on a linear map $V^* \to K$ which is free from uncanonical choices. @Moosbrugger: So $K$ carries the discrete topology? Feel free to add this as an answer :) | |
| Nov 7, 2011 at 18:42 | comment | added | Gerald Edgar | If $V^{**}$ is isomorphic to $W^{**}$, I see no reason to suspect that the isomorphism takes the image of $V$ to the image of $W$. But is that what you mean by "intrinsic"? | |
| Nov 7, 2011 at 18:33 | comment | added | Moosbrugger | My answer was meant as a bit of a joke: certainly one requires hypotheses for such representability theorems. The one I know is called "presentable." But anyway, it's a precise statement for vector spaces for any field $K$. You topologize the dual as a subset of $\prod_{v\in{V}}{K}$, where the latter has the usual ("Tychonoff") topology (and the map $V^*\to\prod_{v\in{V}}{K}$ is $\lambda\mapsto (\lambda(v))_{v\in{V}}$). | |
| Nov 7, 2011 at 15:32 | comment | added | Martin Brandenburg | @Mariano: The annihilator of $V$ is the zero subspace of $V^*$. Perhaps I misunderstand your comment. | |
| Nov 7, 2011 at 15:27 | comment | added | Martin Brandenburg | @Moosbrugger: Not every continuous functor $C^{op} \to \mathrm{Set}$ is representable. For example, $\lim_i \hom(-,x_i)$ is continuous, but is representable iff $\lim_i x_i \in C$ exists. However, there many categories with this property, called SAFT by Theo Johnson-Freyd (mathoverflow.net/questions/49175/…). When is a linear map $V^* \to K$ called continuous? Do you assume that $K$ is a topological field, or even $\mathbb{R}$ or alike? | |
| Nov 7, 2011 at 15:19 | history | edited | Martin Brandenburg | CC BY-SA 3.0 |
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| Nov 7, 2011 at 15:06 | comment | added | Moosbrugger | Continuity is equivalent to representability (in both cases). | |
| Nov 7, 2011 at 15:03 | comment | added | Mariano Suárez-Álvarez | The image is contained, at least, in the annihilator of the annihilator of $V$. | |
| Nov 7, 2011 at 14:56 | history | asked | Martin Brandenburg | CC BY-SA 3.0 |