Hilbert spaces and time series 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 $\mathcal{P}_{\mathcal{M}}Y_{t+1}$?
 A: Question: "Why are future values found by projecting onto the subspace..."
Answer: Because the projection is the conditional expectation of $Y_{t+1}$ given the sigma-field generated by $Y_1,\dots,Y_t$, and it is known in time series analysis that this conditional expectation is, in a specific sense, the best predictor.
Here is a sketch of the geometry behind the first claim:

Take $Y$ as being $Y_{t+1}$, and $\mathscr{G}=\sigma(Y_1,\dots,Y_t)$.
A: If I assume that Cardinal's assumptions are true the reason that the predicted value for Yt+1 in the space M should be the orthogonal projection is because it is the closest point to Yt+1 in M based on the inner product metric.
A: "Prediction" that uses the Hilbert space structure is, as Michael Chernick says above, orthogonal projection of $Y_{t+1}$ onto the subspace generated by the "predictors" $\{ Y_u, u \leq t\}$. 
This is linear regression in the population sense and not the same as orthogonal projection onto $L^2(\sigma(Y_t, Y_{t-1}, \cdots))$, i.e. conditional expectation with respect to $\sigma(Y_t, Y_{t-1}, \cdots)$. The latter subspace is in general much larger than the one generated by $\{ Y_u, u \leq t\}$.  
An explicit calculation is just linear algebra. Geometrically, a Hilbert space is no different from $\mathbb{R}^n$ with the Euclidean inner product (except that it's not locally compact but that's not relevant here).
Denote the autocovariance function of ${Y_t}$ by $\gamma(h)$. One-step ahead prediction means finding $\phi_u$, $u = 1, 2, \cdots$ s.t.
$$
\|Y_{t+1} - \sum_{u \geq 1} \phi_u Y_{t+1-u}\|^2 
$$
is minimized. If $\langle Y_s, Y_s + h \rangle = \gamma(h) = 0$ for all $h > p$, then
$$
\begin{bmatrix}
\gamma(0) & \gamma(1) & \cdots & \gamma(p-1) \\
\gamma(1) & \gamma(0) & \vdots & \gamma(p-2)    \\
\vdots    & \vdots    & \ddots & \vdots \\
\gamma(p-1) & \gamma(p-2) &\cdots &\gamma(0)
\end{bmatrix}
 \begin{bmatrix}
\phi_1  \\
\phi_2 \\
\vdots \\
\phi_p
\end{bmatrix}
=
\begin{bmatrix}
\gamma(1)  \\
\gamma(2) \\
\vdots \\
\gamma(p)
\end{bmatrix}.
$$
Assuming the matrix on the left hand side is positive definite, invert it and you're done. For $n$-step ahead prediction, shift the right hand side forward by $n$.
In the general case where $\gamma$ does not have finite support but, say, is absolutely summable, taking matrices of increasing size gives an approximating sequence.
