Banach space property of parabolic Sobolev space Let $X$ be a real Banach space with norm $||\cdot||$. We define for $1<p<\infty$ and $t_1<t_2$, the space $Y=L^p(t_1,t_2;X)$ to be the space of measurable functions $f:(t_1,t_2)\to X$ such that the norm
$$
||f||_{L^p(t_1,t_2;X)}:=\Big(\int_{t_1}^{t_2}||f||_{X}^p\,dt\Big)^\frac{1}{p}<\infty.
$$
My question is whether the space $Y$ is a reflexive Banach space?
Can you kindly help me.
Any reference is alo very much appreciated.
Thanking you. 

If you use this post, you can show the desired result by applying the result twice provided that X is separable and reflexive.
