Let $p \in [1, + \infty).$ It is well known that $L_p$ spaces are Banach spaces. However, I am wondering whether they are Asplund. What about the case $p = + \infty$?
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It appears that a space is Asplund iff its dual has the Radon-Nikodym property, and this is true (in particular) for all reflexive spaces. So, if you want to show that $L^p$ is Asplund for $p\in (1,\infty)$, then we can use that its dual is $L^q$ (with $1/p+1/q=1$), and $L^q$ is reflexive.
For some references on the RND, see A good resource on the Radon-Nikodym Property in reflexive Banach Spaces?
and
"Geometric Aspects of Convex Sets with the Radon-Nikodým Property" by Bourgin (who also talks about Asplund spaces). I believe you can also find it (or what's needed to prove it) in Dunford and Pettis.
cmk
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1Is it Jean Bourgain or Richard Burgin? – Giuseppe Negro Sep 29 '20 at 23:46
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@GiuseppeNegro thanks for the catch! Slip on my end :) – cmk Sep 29 '20 at 23:53
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1This was subtle, because it appears that Jean Bourgain also has at least one paper on this subject. – Giuseppe Negro Sep 30 '20 at 09:45