Banach space property of parabolic Sobolev space

Question

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.

Answer 1

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