How to prove that Bochner space $L^2(0,T; L^p(\Omega))$ is complete with norm $$\|f\|_{L^2(0,T; L^p(\Omega))} = \left(\int_0^T \|f(t)\|^2_{L^p} dt\right)^{\frac{1}{2}}$$ where $1<p<\infty$?
Attempt: I take Cauchy sequence $(f_n)_n$ in $L^2(0,T; L^p(\Omega))$ and then I get stuck when I try to conclude that $(f_n(t))_n$ is a Cauchy sequence in $L^p(\Omega)$ for every $t\in [0,T]$.
When I think about it, even if I manage to get a limit $f(t) = \lim_n f_n(t)$ for every $t$, I would still have to show that function $f$ is in $L^2([0,T])$, right?
Thank you for your time.
