Timeline for Order density of smooth functions among continuous functions?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Jul 3, 2012 at 5:08 | comment | added | Alex Becker | Yes, they're very useful as a source of counterexamples. Essentially, they show that the derivatives of a function $\mathbb R^n$ on a subset of $\mathbb R^n$ carry only local information, even if that function is smooth. Contrast this with the case of $\mathbb C^n$, where knowing the derivatives of a differentiable function at a single point completely determines the function! | |
| Jul 3, 2012 at 5:03 | comment | added | Leo | Thank you very much for everything, much appreciated. I never had much dealings with bump functions, but now they seem quite useful. | |
| Jul 3, 2012 at 4:57 | comment | added | Alex Becker | @LeonLampret I found a similar fact (that $C^r_b(\mathbb R,\mathbb R)$ is not a lattice) mentioned on page 7 of "Integration: A Functional Approach" by Bichteler, which can be found at books.google.com/…. The desired result follows easily. | |
| Jul 3, 2012 at 4:55 | vote | accept | Leo | ||
| Jul 3, 2012 at 4:55 | comment | added | Leo | Yes, you are right, bump functions are what I really needed. Thank you! May I humbly ask for just one vote up, because I currently can't ask questions (as comments) in other threads? | |
| Jul 3, 2012 at 4:49 | history | edited | Alex Becker | CC BY-SA 3.0 |
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| Jul 3, 2012 at 4:39 | comment | added | Alex Becker | I'm afraid I don't have any references, I just came up with the proof. And $x\vee -x$ is "the least upper bound of $x$ and $-x$", which I showed does not exist. | |
| Jul 3, 2012 at 4:36 | comment | added | Leo | Could you please provide any reference for this theorem? I haven't found this in Duistermaat & Kolk's Multidimensional Real Analysis 1 & 2, nor in Callahan's Advanced Calculus, which are the more advanced calculus books that I have. Should I look among Functional Analysis books? Also, what of the nonexistence of $x\vee -x$ in $\mathcal{C}^r([−1,1],\mathbb{R})$? | |
| Jul 3, 2012 at 4:32 | history | edited | Alex Becker | CC BY-SA 3.0 |
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| Jul 3, 2012 at 4:12 | history | answered | Alex Becker | CC BY-SA 3.0 |