Let $H$ be a Hilbert space and let $V$ be a closed subspace of $H$. Let $x \in H$. Show that there is a unique decomposition $x=u+v$ such that $v \in V$ and $u \in V^{\perp}$.

Now suppose $(\Omega, \mathcal{F}, \mathbb{P})$ is a probability space and let $X \in L^{2}(\Omega, \mathcal{F}, \mathbb{P})$. Suppose $\mathcal{G}$ is a sub- $\sigma$-algebra of $\mathcal{F}$. Define $\mathbb{E}(X \mid \mathcal{G})$ using a decomposition of the above type. Show that $\mathbb{E}\left(\mathbb{E}(X \mid \mathcal{G}) .1_{A}\right)=\mathbb{E}\left(X .1_{A}\right)$ for each set $A \in \mathcal{G}$.

Let $\mathcal{G}_{1} \subseteq \mathcal{G}_{2}$ be two sub- $\sigma$-algebras of $\mathcal{F}$. Show that (a) $\mathbb{E}\left(\mathbb{E}\left(X \mid \mathcal{G}_{1}\right) \mid \mathcal{G}_{2}\right)=\mathbb{E}\left(X \mid \mathcal{G}_{1}\right)$; (b) $\mathbb{E}\left(\mathbb{E}\left(X \mid \mathcal{G}_{2}\right) \mid \mathcal{G}_{1}\right)=\mathbb{E}\left(X \mid \mathcal{G}_{1}\right)$.

No general theorems about projections on Hilbert spaces may be quoted without proof.