Timeline for Extending a homeomorphism from a dense set [closed]
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Oct 13, 2013 at 10:42 | history | closed |
Benoît Kloeckner Tom LaGatta Michael Greinecker Ramiro de la Vega Daniel Moskovich |
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| Oct 13, 2013 at 8:58 | comment | added | fedja | Take the Fourier transform from $L^p$ on the circle to $\ell^q$ ($p<2<q=\frac{p}{p-1}$) then. Looks like you'll need to disclose everything before we run out of stupid counterexamples :-). | |
| Oct 13, 2013 at 8:56 | review | Close votes | |||
| Oct 13, 2013 at 10:42 | |||||
| Oct 13, 2013 at 8:50 | comment | added | Tom LaGatta | (voting to close, thanks everybody) | |
| Oct 13, 2013 at 8:45 | comment | added | Tom LaGatta | @fedja: Let me tell you more of my context, since these examples were not what I was thinking of. I have two Hausdorff linear spaces $X$ and $Y$, and a measurable linear map $f : X \to Y$. In the presence of some additional structure, I have Hilbert subspaces $H_X \subseteq X$ and $H_Y \subseteq Y$, and I've proved that the restriction map $f : H_X \to H_Y$ is an isomorphism. I would like to extend this isomorphism to the closure $\overline{H_X}$ in $X$. I had suspected it might be true for trivial topological reasons (hence this MO question), but it requires something subtler. | |
| Oct 13, 2013 at 8:39 | comment | added | fedja | Something is strange: take $X=Y=\mathbb R$, $D=\mathbb Q$. Put $f(x)=x$ on $D$ and $f(x)=\sqrt 2$ on $X\setminus D$. I'm pretty sure you didn't have this idiotic example in mind, so what conditions are missing? | |
| Oct 13, 2013 at 8:39 | comment | added | Benoît Kloeckner | Even continuity is not enough to deduce homoemorphism: look at the inclusion of a dense interval to the circle. For the second question, look at the inclusion of the rational numbers in the set of real numbers. | |
| Oct 13, 2013 at 8:38 | comment | added | Tom LaGatta | @ToddTrimble: thanks, that's a clean counterexample. | |
| Oct 13, 2013 at 8:36 | comment | added | Todd Trimble | The inclusion $f: (0, 1] \to [0, 1]$ with $D = X$?? | |
| Oct 13, 2013 at 8:24 | history | edited | Tom LaGatta | CC BY-SA 3.0 |
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| Oct 13, 2013 at 8:14 | history | edited | Tom LaGatta | CC BY-SA 3.0 |
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| Oct 13, 2013 at 8:02 | history | asked | Tom LaGatta | CC BY-SA 3.0 |