Timeline for Application of uniform boundedness (Banach-Steinhaus) principle
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Nov 18, 2020 at 16:18 | comment | added | Fozz | Strictly speaking, I think the space $\dot{H^1}$ is defined in terms of equivalence classes where functions that differ by polynomials are identified. In particular, all constant functions are equivalent to the zero function. | |
| Nov 18, 2020 at 16:11 | comment | added | Ayman Moussa | @Fozz what about $u_R=R$ (constant function) which is bounded in $\dot{H}^1$ and for which $\sup_R ||\phi u_R||_{H^1} = \infty$ as soon as $\phi \neq 0$ ? | |
| Nov 18, 2020 at 16:08 | comment | added | Fozz | The statement $\sup_{R}||\phi u_R||_{{H^1}}<\infty$ holds as well because $u_R\in \dot{H^1}$ and $\phi$ has compact support so their product is in $H^1$. | |
| Nov 18, 2020 at 9:30 | comment | added | Hannes | Should it maybe be "$\sup_R \|\phi u_R\|_{\dot{H}^1} < \infty$", so with the homogeneous Sobolev (= gradient) norm? Also, what properties of $\dot{H}^1$ are assumed at the point in the book? For instance, if I see it correctly, if one uses the Sobolev embedding and reflexivity for $\dot{H}^1$, that should do it. (Extract weakly convergent subsequence of $u_R$, use Sobolev embeding, then the weak limit in $\dot{H}^1$ must coincide with $u$ in, say, $L^2_{loc}$.) The UBP argument eludes me, though. | |
| Nov 18, 2020 at 8:24 | history | edited | YCor | CC BY-SA 4.0 |
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| Nov 18, 2020 at 3:11 | answer | added | katago | timeline score: 1 | |
| Nov 18, 2020 at 2:52 | history | edited | Fozz | CC BY-SA 4.0 |
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| Nov 18, 2020 at 2:39 | history | edited | Fozz | CC BY-SA 4.0 |
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| Nov 18, 2020 at 2:23 | history | edited | Fozz | CC BY-SA 4.0 |
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| Nov 18, 2020 at 2:17 | history | asked | Fozz | CC BY-SA 4.0 |