Timeline for An abstract characterisation of weak* topologies
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Dec 15, 2022 at 4:47 | vote | accept | HardyHulley | ||
| Dec 14, 2022 at 20:48 | comment | added | Robert Furber | The bounded weak-* topology is easier to characterize than the weak-* topology, therefore (if you have a locally convex topology on a normed space in which the unit ball is compact, and the topology is the finest linear topology agreeing with the topology on the ball, this space is a dual Banach space with the bounded weak-* topology). Of course, the topology on the unit ball is a piece of structure, not something intrinsic to the normed space structure, because of the existence of Banach spaces with non-isometric preduals. | |
| Dec 14, 2022 at 20:45 | comment | added | Robert Furber | Nik Weaver's answer has an extra assumption that is stated in the question and Jochen Wengenroth's comment (to Nik's answer) - that the topology on $X'$ is also coarser than $\sigma(X',X'')$ (i.e. the weak (not weak-) topology of $X'$). Without this assumption, there is the bounded weak- topology on $X'$ which is strictly finer than $\sigma(X',X)$ in the infinite-dimensional case. In fact it is the finest unique finest linear topology agreeing with $\sigma(X',X)$ on the unit ball. | |
| Dec 14, 2022 at 20:13 | comment | added | Robert Furber | If $X$ is a Banach space equipped with a locally convex topology $\mathcal{T}$ in which $B_X$ is compact, then $X$ is the dual space of the linear maps $X \rightarrow k$ that are continuous when restricted to $B_X$ (where $k$ is the base field $\mathbb{R}$ or \mathbb{C}$). The "locally convex" is necessary by a counterexample of J. W. Roberts. | |
| Dec 14, 2022 at 8:25 | answer | added | Mikael de la Salle | timeline score: 8 | |
| Dec 14, 2022 at 8:20 | comment | added | Jochen Wengenroth | Not meant seriously: Take $\tau_X=\sigma(X,Y)$ if $X$ is the dual of a Banach space $Y$ and whatever topology if $X$ does not have a predual. This nonsense indicates that should should specify further properties, e.g., functoriality of $X \mapsto (X,\tau_X)$. | |
| Dec 14, 2022 at 7:56 | comment | added | HardyHulley | @PietroMajer I definitely don’t want the elements of $X^*$ to be continuous under the topology, because in that case $X$ would have to be finite-dimensional (reflexive AL spaces are finite-dimensional). | |
| Dec 14, 2022 at 7:12 | comment | added | Pietro Majer | As a remark: if such a tvs topology on X leaves all elements of X* continuous, then it is finer than the w tooology of X, $\sigma(X,X^*)$ . So the ball of X is also w compact, and X is reflexive by kakutani thm. | |
| Dec 14, 2022 at 6:46 | history | edited | Martin Sleziak |
added a top-level tag; see: https://meta.mathoverflow.net/questions/1457/why-are-mo-tags-formatted-as-they-are
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| Dec 14, 2022 at 6:46 | comment | added | Martin Sleziak | On Mathematics: An abstract characterisation of weak* topologies | |
| Dec 14, 2022 at 6:01 | history | asked | HardyHulley | CC BY-SA 4.0 |