Timeline for Whether Krein-Milman property implies Radon-Nikodym property
Current License: CC BY-SA 4.0
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| Apr 4, 2023 at 11:15 | comment | added | Tony Prochazka | For what is worth, these properties coincide also in Lipschitz-free spaces, but it is for stronger reason (failure of the RNP already implies the presence of $L_1$). It is proved in Purely 1-unrectifiable metric spaces and locally flat Lipschitz functions. Trans. Amer. Math. Soc. 375 (2022), no. 5, 3529–3567. | |
| Nov 23, 2019 at 13:27 | comment | added | Tanmoy Paul | I would like to add one comment and one question in the same thread. RNP is closely related to Asplund property. A Banach space $X$ is said to be an Asplund space if every continuous convex function is Frechet differentiable in a dense $G_\delta$ set. Again it has many equivalent characterizations like $X$ is Asplund if and only if every equivalent norm on it has at least one point of Frechet differentiability also, $X$ is Asplund iff $X^*$ have RNP. It is known that if $X$ has an equivalent norm which is Frechet diff then $X$ is Asplund. My question is whether the converse is also true? | |
| Nov 19, 2019 at 8:22 | history | edited | Tomasz Kania | CC BY-SA 4.0 |
added 6 characters in body
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| Nov 18, 2019 at 14:39 | history | answered | Tomasz Kania | CC BY-SA 4.0 |