I'm having a hard time understanding the difference between Linear Least Square Estimate vs. Minimum Mean Squares Estimate. I can see that their equations are different, but from a geometric standpoint I'm a little bit confuse.

So letting X, Y be random vectors of dimension m and n respectively and finite means/variances, we have that

(Linear Least Square Estimate) L[XY]=E[X]+cov(X,Y)Y1(YE(Y))

(Minimum Least Square Estimate) MMSE[XY]=E[XY]

I understand that the LLSE is the projection of X onto the space of all linear functions of Y and I also understand that MMSE is the projection of X onto the space of all functions of Y. I can see that the space L(Y) which consists of all linear functions of Y is a subspace of the space G(Y) which consists of all functions of Y. I understand that the projection X onto G(Y) isn't necessarily the same as the projection of X onto L(Y). That being said E[XY] is a linear function of Y, so why isn't it the case that L[XY]=MMSE[XY] regardless of X and Y since MMSE[XY]L(Y)?