Timeline for Compactness of the unit ball of a Banach space for topologies finer than the weak* topology
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Jul 31, 2022 at 9:55 | vote | accept | Goulifet | ||
| Jul 30, 2022 at 14:18 | comment | added | Christian Remling | @GeraldEdgar: Yes, this is right; compare Michael's comment above. We are looking for stronger (than weak $*$) topologies on $X^*$ that induce the same topology as before on balls. | |
| Jul 30, 2022 at 1:21 | comment | added | Gerald Edgar | Isn't this true: in any compact Hausdorff space, any finer topology is non-compact; and any courser topology is non-Hausdorff? Equivalently: a continuous map from one compact Hausdorff space to another is automatically a homeomorphism. | |
| Jul 29, 2022 at 19:20 | comment | added | Michael Greinecker | @Goulifet Yes, that is correct. | |
| Jul 29, 2022 at 19:07 | comment | added | Goulifet | @MichaelGreinecker this is an interesting statement. But then if I have another topology $\tau$ finer than the weak*-topology $\tau_*$, I can deduce that the traces of $\tau$ and $\tau_*$ on the unit ball coincide. But not necessarily that $\tau$ and $\tau_*$ coincide as topologies on $\mathcal{X}*$, right? | |
| Jul 29, 2022 at 18:55 | comment | added | Goulifet | @ChristianRemling: you are right, we actually want a vector space topology that makes, for instance, continuous and functional via elements of $\mathcal{X}$ plus possibly one additional functional via an element of $\mathcal{X}'' \backslash \mathcal{X}$$ (as in Nik Weaver approach, see below). | |
| Jul 28, 2022 at 17:27 | comment | added | Michael Greinecker | If you look just at the trace of the unit ball, there is not. Any two comparable compact Hausdorff topologies on a set coincide. | |
| Jul 28, 2022 at 17:26 | comment | added | ottone | Perhaps relevant: it is a well known classical Banach space result that the finest topology (not necessarily lc or linear) on a dual Banach space which agrees with the weak star topology on balls is the Mackey topology (uniform convergence on weak compacta). If one assumes linearity, then one only need look at the unit ball. | |
| Jul 28, 2022 at 14:57 | comment | added | Christian Remling | So perhaps you really want vector space topologies? | |
| Jul 28, 2022 at 14:57 | comment | added | Christian Remling | You can just take any $x'\in X'$, $\|x'\|>1$, and let the weak $*$ open sets together with $\{ x'\}$ generate a stronger topology. This still induces the same topology on $B$ as before. | |
| Jul 28, 2022 at 14:06 | history | edited | Jochen Wengenroth |
edited tags
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| Jul 28, 2022 at 13:35 | answer | added | Nik Weaver | timeline score: 10 | |
| Jul 28, 2022 at 13:06 | history | edited | Goulifet | CC BY-SA 4.0 |
question has been precised
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| Jul 28, 2022 at 12:58 | history | asked | Goulifet | CC BY-SA 4.0 |