Timeline for The topology of $C_0^\infty(M) $
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| May 12, 2012 at 21:57 | comment | added | Marc Palm | Damn, $| f ||_{\alpha,i} = sup_{x \in U_i} | \partial_\alpha f(x) \phi_i(x)|$. Will this be okay? | |
| May 12, 2012 at 21:55 | comment | added | Marc Palm | Thank you for the feedback. Probably, I read the questions in the way, I face questions when experiencing the question. I would guesss, that the final definition should envolve a variant of the partition. Here is a first suggestion: Fix a relative compact cover $(U_i)_i$, where $U_i$ is isomorphic to an open disc of $\mathbb{R}^n$ and functions $\phi_i$, which are constant one on $U_i$. Define for a multi-index $\alpha \in \mathbb{N}_0^n$ and define $|| f ||_{\alpha, i} = sup_{x \in U_i} |f(x) \phi_i(x)|$ and define $|| f ||_k = \sup_{\alpha_1 + \dots +\alpha_n=k,i} || f ||_{\alpha, i}$. | |
| May 12, 2012 at 21:06 | comment | added | Deane Yang | It would be nice if someone could write out the details involving a set of co-ordinate charts and the corresponding partition of unity. | |
| May 12, 2012 at 20:50 | comment | added | paul garrett | And, if this is the issue, one probably be clear that the set of charts is maximal. Sometimes this is implicit in "atlas", but sometimes not. As the questioner may not already know, there is at least a linguistic hitch in subsequent discussions that begin "take a local coordinate map/patch/whatever so that..." if the original legal collection of such things is not chosen to be maximal (self-consistent, of course). This matters less if one does not attempt to specify "charts" completely at the beginning, but definitions vary... | |
| May 12, 2012 at 20:32 | comment | added | Deane Yang | The person asking says he or she already understands the topology when $X$ is equal to $R^n$ but not when $X$ is a manifold. Aren't the partition of unity and a choice of co-ordinate charts (just one atlas is enough, no?) the key points of an answer to the question? | |
| May 12, 2012 at 20:21 | history | edited | Marc Palm | CC BY-SA 3.0 |
added 220 characters in body
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| May 12, 2012 at 20:17 | comment | added | Marc Palm | I see now: if we require a smooth partition of unity, for the metrizabilty issue the partition will probably have to be countable to generalize to general paracompact smooth manifold. Is this your point, or am I overseeing something. | |
| May 12, 2012 at 20:13 | history | edited | Marc Palm | CC BY-SA 3.0 |
added 58 characters in body
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| May 12, 2012 at 20:10 | comment | added | Marc Palm | Okay, there I was sloppy;) | |
| May 12, 2012 at 16:00 | comment | added | Deane Yang | But what's the definition of $f'$ when $X$ is a manifold? | |
| May 12, 2012 at 15:07 | history | edited | Marc Palm | CC BY-SA 3.0 |
added 143 characters in body; added 5 characters in body
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| May 12, 2012 at 14:55 | history | edited | paul garrett | CC BY-SA 3.0 |
Dropped "dual" on test functions in last line
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| May 12, 2012 at 13:28 | history | answered | Marc Palm | CC BY-SA 3.0 |