In Casella&Berger, there is a theorem as follows:
Theorem 7.3.20 If $E_\theta W = \tau(\theta)$, $W$ is the best unbiased estimator of $\tau(B)$ if and only if $W$ is uncorrelated with all unbiased estimators of $0$.
If we assume $E(X)=0$ means X is orthogonal to the vector of $(1,1,1,1,...)$, then "uncorrelated with all orthogonal vectors to $(1,1,1,1,...)$" means being in the vector space of $(1,1,1,1,...)$. Thus, any such estimator, $W$, must be constant for all $\textbf{x}$.
What is wrong here?
EDIT:
The assumption I made comes from this text, Introduction to probability - Blitzstein
:
We can think of unconditional expectation as a projection too: $E(Y)$ can be thought of as $E(Y|0)$, the projection of $Y$ onto the space of all constants.
If such assumption is false, can you elaborate what is this text trying to say and what is the correct interpretation?
EDIT: Please give feedback in words. I brought two statements from two famous books, and a contradicting result which comes from these two statements with detailed reasoning. How can I add details or clarity to this question?