Predictable Projection in is a Hilbert Space Projection?

Question

Recently I began reading about predictable projections of stochastic processes. Suppose that $X_t$ is a $(F_t)$-adapated stochastic process defines on a stochastic base $(\Omega,(F_t)_{0\leq t\leq T},\mathbb{P})$ and let $m$ be the Lebesgue measure on $\mathbb{R}$. Suppose that $X_t$ satisfies the $L^2$-integrability condition: $$ E\left[ \int_0^T |X_t|^2 dt \right]<\infty $$ i.e.: $X_T$ is $L^2_{P\otimes m}(F_T\otimes \Omega) $ then can the predictable projection of $X_t$ onto the predictable filtration be understood as a Hilbert space projection of $X_T$ onto $L^2_{P\otimes m}(Pred)$; where $Pred$ is the predictable $\sigma$-algebra?