Suppose $(\Omega,\mathcal F,\mathbb P)$ is a complete probability space, on which a filtration is defined. Let $P$ be the set of progressively measurable processes $X$ such that $\Vert X\Vert_P^2\equiv\mathbb E[\sup_{0\leq t\leq T}|X_t|^2]<\infty$. With two processes being identified if and only if they have almost surely equal paths, is $(P,\Vert\cdot\Vert_P)$ a Banach space?
I try to interpret this as $L^2$ over some function space $A$, but I have difficulty describing this $A$ precisely to capture the progressive measurability. Also, supremum is used here rather than essential supremum, so I'm not sure if $A$ can be described as some $L^\infty$, However, because indistinguishility is used to identify processes here, it looks like using supremum is enough. Sorry I'm a bit lost.
Could someone give me a proof or disproof, thanks!