Suppose that $\{Y_{t}: t \in \mathbb{Z} \}$ is a stationary zero mean time series. Consider the Hilbert space $\mathcal{H}$ generated by the random variables $\{Y_t: t \in \mathbb{Z} \}$ with inner product $ \langle X, Y \rangle = E(XY)$ and norm $||X||^2 = E|X|^2$
Consider the subspace $\mathcal{M}$ generated by the random variables $\{Y_u: u \leq t \}$. Why are future values found by projecting onto the subspace $\mathcal{M}$? For example, why is $Y_{t+1}$ found by $\mathcal{P}_{\mathcal{M}}Y_{t+1}$?