Timeline for Tensor product of certain Sobolev spaces on non-compact manifolds
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Aug 31, 2014 at 11:17 | vote | accept | AlexE | ||
| Aug 31, 2014 at 10:24 | history | edited | Peter Michor | CC BY-SA 3.0 |
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| Aug 30, 2014 at 13:07 | comment | added | AlexE | Are my five points all correct? I also have two remaining questions: (1) What is a reference for the fact that $C_c^\infty(M) \otimes_{\mathrm{alg}} C_c^\infty(N) \hookrightarrow C_c^\infty(M \times N)$ is dense in the "usual" topology? This probably just boils down to a reference for the fact that the projective tensor product respects direct limits of locally convex spaces. (2) Since "my" topology does not make the spaces nuclear, what is the difference if we choose another tensor product in my question instead of the projective one? Thanks in advance and for the already given answer! | |
| Aug 30, 2014 at 13:01 | comment | added | AlexE | (iv) Passing to the completions, we may conclude that the answer to my second question is "yes", i.e., $W^{\infty,1}(M) \ \hat{\otimes} \ W^{\infty,1}(N) \cong W^{\infty,1}(M \times N)$. (v) The space $s$ you talk about is the same space $s$ as in this answer of you: mathoverflow.net/q/173588/13356. | |
| Aug 30, 2014 at 12:57 | comment | added | AlexE | I will summarize your answer in order to make sure that I understood it correctly: (i) The topology on $C_c^\infty(M)$ that you talk about in the first paragraph of your edit is the "usual" one that one considers on this space (i.e., uniform convergence of all derivatives on all compacta). (ii) This topology is finer than the one that I consider (the one induced from $W^{\infty,1}(M)$). (iii) Since the inclusion $C_c^\infty(M) \otimes_{\mathrm{alg}} C_c^\infty(N) \hookrightarrow C_c^\infty(M \times N)$ is dense in the "usual" topology, it is also dense in "my" topology. | |
| Aug 29, 2014 at 15:13 | history | edited | Peter Michor | CC BY-SA 3.0 |
added more information.
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| Aug 29, 2014 at 10:02 | history | edited | Peter Michor | CC BY-SA 3.0 |
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| Aug 29, 2014 at 7:32 | comment | added | AlexE | And could you please elaborate a bit on how to prove the denseness result in my first question? Thanks. | |
| Aug 29, 2014 at 7:31 | comment | added | AlexE | If the left hand side is dense in the right hand side in my first question, doesn't this imply that the answer to my second question is "yes", because in my second question we just pass to the completions? | |
| Aug 28, 2014 at 18:16 | history | answered | Peter Michor | CC BY-SA 3.0 |