How to prove the UMVUE under these conditions

Hi everyone, the question is

Let $\displaystyle \mathbb{U}$ be the set of all unbiased estimators of 0. Namely,

$\displaystyle \mathbb{U}$={$\displaystyle U(X): E_{\theta}(U(X)) = 0, $ for all $\displaystyle \theta$∈$\displaystyle \Omega$}.

Then prove$\displaystyle \delta(X)$ is an UMVUE of $\displaystyle \theta$ iff for any $\displaystyle U(X)$∈$\displaystyle \mathbb{U}$, $\displaystyle Cov_{\theta}(\delta(X),(U(X)) = 0$.

, and give a geometrical interpretation.

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I only figured that $\displaystyle E_{\theta}(U(X)\delta(X))=0$ so far, and I have no idea if I can use Lemann-Scheffe theorem to prove.

Please help me, and give some tips for me. Thx a lot!