# Some questions on parabolic function spaces

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?

1. This is true if and only if $X$ has the Radon-Nikodym property, see Diestel & Uhl: Vector measures, 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 $L^p(0,T;B)$.
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.