Timeline for Geometrical meaning of spaces that possess the weak* uniform Kadec-Klee property
Current License: CC BY-SA 4.0
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| Aug 25, 2022 at 12:36 | comment | added | Jack L. | .... or if $X$ is a uniformly smooth space (say $\ell^2$), then $X^*$ has UKK. But $\ell^\infty$, as the dual of $\ell^1$ does not have UKK. The Hardy space, $H^1$ and several classical non-reflexive spaces have the w*-UKK. | |
| Aug 25, 2022 at 12:35 | comment | added | Jack L. | .... Indeed, (w*-)UKK is a variation of a topological theme: that of the coincidence of the sequential weak(-star) topology and norm topology in the unit sphere (also known as Kadec-Klee property). Thus, it will appear infeasible to obtain a geometrical characterization as such, beyond that given by the definition .$$~$$ As requested, concerning examples, because Schur spaces (i.e. spaces where the sequential weak and norm topologies agree in the unit ball) and uniformly convex spaces have the UKK, it follows that if $X$ is a predual of $\ell^1$ (say $c_0$) or | |
| Aug 25, 2022 at 12:26 | comment | added | Jack L. | The definition, as you have given above, is the uniform Kadec Klee (simply, UKK) property for the dual space, rather than the weak* uniform Kadec Klee (simply, w*-UKK), which requires weak-star convergence. That said, both properties are equivalent in Grothendieck spaces (i.e. spaces where weak and weak-star convergence coincide)..$$~$$That said, it is properly safe to say that (w*-)UKK is a topological, rather than geometrical, property of the unit sphere.......(cont’d below). | |
| S Aug 24, 2022 at 6:55 | history | suggested | Angel Peñaflor | CC BY-SA 4.0 |
correted spelling, fixed grammar, improved formatting
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| Aug 24, 2022 at 2:10 | review | Suggested edits | |||
| S Aug 24, 2022 at 6:55 | |||||
| Aug 24, 2022 at 1:59 | history | edited | Tomás Pérez Fernández | CC BY-SA 4.0 |
added 4 characters in body
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| S Aug 24, 2022 at 1:48 | review | First questions | |||
| Aug 24, 2022 at 11:17 | |||||
| S Aug 24, 2022 at 1:48 | history | asked | Tomás Pérez Fernández | CC BY-SA 4.0 |