I am stuck on this exercise defining conditional expectation as projection in a Banach space.
Let $X \in L^2([0,1],\mathcal{B}(([0,1])),\mathcal{L}^1)$ and let $\mathcal{A}=\sigma(\{[0,\frac12]\})$. Then we can make $L^2(\Omega,\mathcal{A},\mu)$ a Banach space using a quotient argument. Define $E[X|\mathcal{A}]$ as the $L^2(\Omega,\mathcal{A},\mu)$-orthogonal projection onto $L^2(\Omega,\mathcal{F},\mu)$ for $\mathcal{F}\subset \mathcal{A}$ a sub-$\sigma$-algebra. Give a general formula for $E[X|\mathcal{A}]$.
I am not sure what the formula should be in terms of. In the previous exercise it was established that $E(XY)=E[YE[X|\mathcal{F}]]$ for $X \in L^2(\Omega,\mathcal{A},\mu), Y\in L^2(\Omega,\mathcal{F},\mu)$ so $E(XY)=E(YE(X/\mathcal{A})$. I was thinking investigating how $E(X|\mathcal{A})$ it acts on the smallest components of $\mathcal{A}$, putting $Y=I_{[0,1/2]}$ yields $E(XI_{[0,1/2]})=E(I_{[0,1/2]}E(X|\mathcal{A}))$ but this isn't useful.