Why $\mathbb E[X|Y]$ not equal to $\frac{\mathbb E[XY]}{\mathbb E[Y^2]}Y$ al the time?

Question

I know that if $(X,Y)$ is au gaussian vector, then $$\mathbb E[X|Y]=\frac{\mathbb E[XY]}{\mathbb E[Y^2]}Y.$$ Since $(X,Y)\longmapsto \mathbb E[XY]$ is a scalar product and that $\mathbb E[X|Y]$ can be seen at the orthogonal projection of $X$ on the vector space of $\sigma (Y)-$measurable function, We should have that $$\mathbb E[X|Y]=\frac{\left<X,Y\right>}{\|Y\|^2}Y=\frac{\mathbb E[XY]}{\mathbb E[Y^2]}Y,$$ should be always correct (were $\left<X,Y\right>:=\mathbb E[XY]$ and $\|Y\|=\sqrt{\left<Y,Y\right>}$). But my teacher said that it's not, and I don't understand why.

Answer 1

The definition of the orthogonal projection of $u$ on $W=span\{w_1,...,w_n\}$ where $w_i$ are orthogonal is indeed $$Proj_W(u)=\sum_{i=1}^n\frac{\left<u,w_i\right>}{\|w_i\|^2}w_i.$$

What would be the orthogonal basis of the space of $\sigma (Y)-$measurable function ? If $Y$ would be such a basis, then your formulae would hold. Unfortunately, it's not the case.