Let $X_{t}$ be a stochastic process. For the realizations or random variables the natural metric is the standard deviation and the covariance as the inner product. For simplicity sake, lets assume that $\mathbb{E}X_{t}=0$ and $\mathbb{E}(X_{t}X_{t-j})= \sigma(j)$. One can generate a Hilbert space $H_{t}$ around the sequence of the realizations till period t of such stochastic process i.e. by taking the elements, linear combinations of these as well as the linear combinations of the limits, etc.
By the Hilbert Space Decomposition Theorem:
Let $H_{t} = H_{t-1}\otimes Ct$
where $Ct$ is another Hilbert space orthogonal to $H_{t-1}$. Its dimension is either zero or 1. In the case its zero, then $x_{t} \in H_{t-1}$.
How can I show that
$$\mathbb{V}(x_{t} - \mathbb{P}(x_{t}|H_{t-1}))= 0 $$? where $\mathbb{P}$ denotes projection of $x_{t}$ into $H_{t-1}$.
My approach was the following:
$$\mathbb{E}((x_{t} - \mathbb{P}(x_{t}|H_{t-1} ) - \mathbb{E}(x_{t} - \mathbb{P}(x_{t}|H_{t-1}))^{2}) $$
$$=\mathbb{E}((x_{t} - \mathbb{P}(x_{t}|H_{t-1} ) + \mathbb{E}(\mathbb{P}(x_{t}|H_{t-1}))^{2}) $$ ... by linearity of expectation and the assumption that $\mathbb{E}X_{t}=0$
How should I understand the expectation of a projection to get things to cancel out?