Timeline for A minimality problem for a class of Banach spaces
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Feb 8, 2020 at 22:22 | comment | added | Bunyamin Sari | @Ben W Bill is answering the natural follow up question of what happens if you restrict to $p<2$ and $p>2$. | |
| Feb 8, 2020 at 21:57 | comment | added | Ben W | Did I misunderstand the OP's question? It sounded like he was asking whether every Banach space is a Hilbert space that embeds isomorphically into $(\oplus\ell_p^n)_{\ell_2}$, $p\in[1,2)\cup(2,\infty]$. But the space itself is not isomorphic to a Hilbert space, so the answer is no. | |
| Feb 8, 2020 at 17:26 | comment | added | Bill Johnson | Proposition 21.6 and Remark 2 following In Tomczak-Jaegermann's book "Banach-Mazur Distances..." yield that for for all $2<s<p<\infty$, $\ell_s$ is isomorphic to a subspace of $L_2(\ell_p)$, with uniformly bounded isomorphism constant for $s<(2+p)/2$. It follows that if $s_n\downarrow2$ and $m_n \to \infty$ quickly, then $(\sum_{n=1}^\infty \ell_{s_n}^{m_n})_2$ embeds into $X_{p,2}$ for all $p>2$ but is not isomorphic to a Hilbert space. | |
| Feb 7, 2020 at 15:04 | comment | added | Bill Johnson | OTOH, if $X$ embeds into $X_{p,2}$ for all $p<2$, then $X$ need not be isomorphic to a Hilbert space. Take $p_n \uparrow 2$ and $m_n \to \infty$. Then $(\sum_{n=1}^\infty \ell_{p_n}^{m_n})_2$ embeds into $X_{p,2}$ for all $p<2$ but is not isomorphic to a Hilbert space if $m_n \to \infty$ sufficiently quickly. | |
| Feb 7, 2020 at 14:54 | comment | added | Bill Johnson | Yes; you only need embedding for one $\infty>p>2$ and one $p<2$ to apply Kwapien's theorem. If you have embedding for all $p>2$ and uniform control on the isomorphism constants, then again you get a positive answer (use the fact that any $n$ dimensional subspace of $L_p$ is $n^{|1/p-1/2|}$ isomorphic to $\ell_p^n$). | |
| Feb 6, 2020 at 18:00 | comment | added | N. de Rancourt | I would, then, be interested in the answer to the original question when we restrict our attention to $p>2$. | |
| Feb 6, 2020 at 17:10 | review | Low quality posts | |||
| Feb 6, 2020 at 18:00 | |||||
| Feb 6, 2020 at 16:54 | history | answered | Bunyamin Sari | CC BY-SA 4.0 |