Timeline for interiors of positive cones in ordered Banach spaces
Current License: CC BY-SA 4.0
11 events
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| May 23 at 16:01 | comment | added | Oliver Díaz | Thank you for your explanation. The simple one is the one I was interested the most. I have heard arguments similar to your second explanation based on deeper results about Banach lattices and it was nice of you to present one such explanation too. | |
| May 23 at 6:48 | history | edited | Jochen Glueck | CC BY-SA 4.0 |
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| May 23 at 6:34 | comment | added | Jochen Glueck | Alternatively, an abstract non-sense proof (and a bit more general): Let $E$ be a Banach lattice with order continuous norm such that $E_+$ has non-empty interior. Due to the non-empty interior of $E_+$, $E$ can be equivalently renormed to be an AM-space with unit; thus, $E$ has the Dunford-Pettis property. As $E$ has order continuous norm, all order intervals in $E$ are weakly compact and since $E_+$ has an interior point it follows that the unit ball in $E$ is weakly compact, so $E$ is reflexive. The Dunford-Pettis property plus the reflexivity implies that $E$ is finite-dimensional. | |
| May 23 at 6:33 | comment | added | Jochen Glueck | @OliverDíaz: Yes, sure. Direct proof: Let $p < \infty$. If $L^p$ is infinite-dimensional, one can construct a sequence $(\Omega_n)$ of pairwise disjoint measurable sets of finite non-zero measure. Now let $f \in L^p_+$ and $\varepsilon > 0$. It follows from the monotone convergence theorem that $\|f 1_{\Omega_n}\| < \varepsilon$ for at least one $n$. Let $g$ be equal to $f$ on the complement of $\Omega_n$ and equal to a sufficiently small negative number on $\Omega_n$. Then $g$ is closer than, say, $2\varepsilon$ to $f$, but $g$ is not in $L^p_+$. So $f$ is not an interior point of $L^p_+$. | |
| May 23 at 2:44 | comment | added | Oliver Díaz | do you know if a simple proof that $L^+_p(\mu)$ has empty interior iff $L_p(\mu))$ has infinite dimension? or a reference perhaps? | |
| Nov 7, 2023 at 20:53 | history | edited | Jochen Glueck | CC BY-SA 4.0 |
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| Nov 7, 2023 at 20:44 | history | edited | Jochen Glueck | CC BY-SA 4.0 |
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| Oct 28, 2023 at 12:56 | history | edited | Jochen Glueck | CC BY-SA 4.0 |
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| Oct 28, 2023 at 11:05 | vote | accept | Saito | ||
| Oct 28, 2023 at 5:31 | history | edited | Tobias Fritz | CC BY-SA 4.0 |
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| Oct 27, 2023 at 19:18 | history | answered | Jochen Glueck | CC BY-SA 4.0 |