If $\mathbb E[X\mid Y]$ can be seen as a projection why $\mathbb E[X\mid Y]=\frac{\mathbb E[XY]}{\mathbb E[Y^2]}Y$ is not always true? We know that $\mathbb E[XY]$ is a scalaire product on $L^2(\mathbb P)$. In a book (an introduction to stochastic differential equation of Evans) page 30-31, it's written that $\mathbb E[X\mid Y]$ can be seen as the orthogonal projection of $X$ on $Y$. So, why $$\mathbb E[X\mid Y]=\frac{\mathbb E[XY]}{\mathbb E[Y^2]}Y,$$
not always true ? 

Well if you take $X$ and $Y$ to be independent, the equation reads $$1 = \frac{Y\mathbb{E}(Y)}{\mathbb{E}(Y^2)},$$ which seems a little bit weird.
