I remember I read those problems some place, but I cannot find it. Does anyone have any idea where I can find it?


1. If $X$ is a Banach space, then $(L^1(a,b;X))^\*\cong L^\infty(a,b;X^\*)$?
2. $X, Y$ are both Banach spaces, if $X$ is compactly embedded into $Y$, then do we have $L^p(a,b;X)$ is embedded in $L^p(a,b;Y)$ compactly?
3. If $X\_1$, $X\_2$ are both reflexive Banach spaces, will $(X\_1, X\_2)\_\theta$ be reflexive?



Some questions on parabolic function spaces
1. This is true if and only if $X$ has the Radon-Nikodym property, see Diestel & Uhl: [Vector measures](http://bookstore.ams.org/surv-15), Chapter IV.1, Theorem 1.
2. This would be very useful indeed, but unfortunately, no. The standard reference to get started is probably the paper by Jacques Simon, [Compact sets in the space $L^p(0,T;B)$](https://link.springer.com/article/10.1007/BF01762360).
3. This depends on the interpolation functor $(\cdot,\cdot)\_{\theta}$, but you will in general need more compatibility between $X\_1$ and $X\_2$ such as their intersection being dense in both spaces, and the same for their duals. See e.g. Chapters 1.11.2 and 1.11.3 in Triebel's [Interpolation Theory, Function Spaces, Differential Operators](https://books.google.at/books/about/Interpolation_Theory_Function_Spaces_Dif.html?id=HzvvAAAAMAAJ).


