Consider the probability space $(\Omega,\mathcal{A},P)$ and a sub-$\sigma$-algebra $\mathcal{G}\subseteq\mathcal{A}$. We know that $L^2(P)=L^2(\Omega,\mathcal{A},P)$ is a complete normed space by the Fischer-Riesz Theorem. As the norm is induced by the $L^2$-inner product, $L^2(P)$ is understood as a Hilbert space.
(a) Define $H_\mathcal{G}:=L^2(\Omega,\mathcal{G},P)$ and view it as a subset of $L^2(P)$. Prove that $H_\mathcal{G}$ is a nontrivial, closed (linear) subspace of $L^2(P)$.
(b) With part (a), the (Hilbert space) Projection Theorem from the introductory course on functional analysis implies that there exists a unique linear (even continuous) mapping ${\rm pr}_\mathcal{G}:L^2(P)\to L^2(P)$ such that for all $X\in L^2(P)$ and $Y\in H_\mathcal{G}$ holds $${\rm pr}_\mathcal{G}(X)\in H_\mathcal{G}\quad\text{and}\quad(({\rm Id}-{\rm pr}_\mathcal{G})(X),Y)_{L^2(P)}=E[(X-{\rm pr}_\mathcal{G}(X))\cdot Y]=0.$$ Prove that ${\rm pr}_\mathcal{G}(X)=E[X\mid\mathcal{G}]$ for all $X\in L^2(P)$.
ref: [Sa19] Jörn Saß, Probability Theory cf. [Sa19, Appendix A.22 part (ii)].