Timeline for finite dimensionality of a subspace of a Banach space
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Oct 30, 2023 at 13:41 | comment | added | Mikael de la Salle | @T.Le Yes, the factor $x^2$ is precisely what what the construction possible. Let $f$ be any compactly supported function on $(0,1)$ such that $\int |f|^2 = \int x^2 |f'_n|^2 = 1$. Then the same is true for the function $\varepsilon^{-1/2} f(\cdot/\varepsilon)$, and if the sequence $\varepsilon(n)$ decreases fast enough to $0$, then the functions $v_n=f_{\varepsilon(n)}$ have disjoint support, contained in $(0,\delta)$. | |
| Oct 28, 2023 at 13:20 | comment | added | T. Le | @MikaeldelaSalle Is there an easy way to see the existence of the sequence $\{v_n\}$? On the other hand, without the factor of $x^2$ in the integral with $v'_n$, I believe there is no such sequence due to the fact that for a bump function $v$ supported on an interval $I$, $\|v\|_{L^2}\leq \sqrt{|I|}\|v'\|_{L^2}$. | |
| Oct 28, 2023 at 13:08 | comment | added | T. Le | @Ali, I don't know whether the linear combinations of my functions also satisfy the same condition. That was what I missed when thinking about the problem! | |
| Oct 28, 2023 at 3:07 | comment | added | Mikael de la Salle | No. Let $v_n$ be a sequence of functions supported in $(0, \delta)$, with disjoint support and such that $\int |v_n|^2 dx = \int x^2 |v'_n|^2 dx=1$ (carefully chosen bump functions). Let $w_n$ be any sequence supported in $(\delta,1)$ in the unit ball of $H$. Let finally $u_n=v_n + 2^{-n} w_n$. Then the linear span $W$ of the $u_n$'s satisfy your condition, but $W\left|_{(\delta,1)}\right.$ is the span of the $w_n$'s, so it can be any countable dimension subspace. | |
| Oct 28, 2023 at 0:36 | comment | added | Ali | But do all of their linear combinations also satisfy the same bound? | |
| Oct 27, 2023 at 23:59 | comment | added | T. Le | I'm not sure whether I understood your question correctly but it seems to me that for a given $C>0$, all the functions $x^{-\alpha}$ with $\alpha < \min\{\frac{1}{2},\sqrt{C}\}$ belong to $W$. These functions, when restricted to any interval, are still linearly independent. | |
| Oct 27, 2023 at 20:40 | history | asked | Ali | CC BY-SA 4.0 |