Timeline for Is every sequentially $\sigma(E',E)$-continuous linear functional on a dual Banach space $E'$ necessarily a point evaluation?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 11, 2021 at 19:07 | vote | accept | Ruy | ||
| Jan 11, 2021 at 8:36 | comment | added | Robert Furber | Side note: These questions can bump into set-theoretic undecidability. Consider $\ell^\infty(\mathbb{R})$ (the bounded, not required to be measurable, functions $\mathbb{R} \rightarrow \mathbb{R}$) as the dual space of $\ell^1(\mathbb{R})$. There is a sequentially weak-* continuous functional that is not weak-* continuous (i.e. a point evaluation) if and only if there is an atomless real-valued measurable cardinal (i.e. $(\mathbb{R}, \mathcal{P}(\mathbb{R}))$ admits a probability measure vanishing on singletons). | |
| Jan 11, 2021 at 7:57 | comment | added | Robert Furber | Related: mathoverflow.net/q/284276/61785 | |
| Jan 11, 2021 at 6:35 | answer | added | Nate Eldredge | timeline score: 7 | |
| Jan 10, 2021 at 23:59 | comment | added | Ruy | @Mikael, here is what I was able to do based on your comment: math.stackexchange.com/questions/3977058/… Thanks again! | |
| Jan 10, 2021 at 20:03 | comment | added | Mikael de la Salle | Indeed, I had missed this Corollary. Note however that separability of $E$ is assumed, and used essentially in the proof. | |
| Jan 10, 2021 at 19:58 | comment | added | Ruy | @Mikael, thanks for pointing it out. In fact the following Corollary in Conway's book, namely V.12.8, is exactly what I am asking! | |
| Jan 10, 2021 at 19:42 | comment | added | Mikael de la Salle | This is at least true if $E$ is separable. Indeed, in that case, by the Krein-Smulian theorem, a convex subset of $E'$ is $\sigma(E',E)$-closed if and only if it is $\sigma(E',E)$ sequentially closed. See Corollary V.12.7 in Conway's course in functional analysis. Apply this to the kernel of $\varphi$. | |
| Jan 10, 2021 at 18:09 | history | asked | Ruy | CC BY-SA 4.0 |