Let $H$ be a real separable Hilbert space. An isonormal Gaussian process over $H$ is a centered Gaussian family $W = \{W(h):h \in H\}$ such that $$\mathbb{E}[W(h), W(h^\prime)] = \langle h, h^\prime\rangle_H\ \forall\ h, h^\prime \in H.$$ Let $\tilde{H}$ be a proper real Hilbert subspace of $H$. The $\tilde{H}$-pinned process associated with $W$ is the centered Gaussian family $\{W(\operatorname{proj}(h \mid \tilde{H}^\bot)): h \in H\}$.
How can I prove that $A = \sigma(W(h):h \in \tilde{H})$ and $B = \sigma(W(h):h \in \tilde{H}^\bot)$ are independent and deduce from this that $\mathbb{E}[W(h) \mid B] = W(\operatorname{proj}(h \mid \tilde{H}^\bot))\ \forall\ h \in H$?
You need to prove that $W(h)$ and $W(h')$ are independent, for each pair $h\in\bar{H}$ and $h'\in\bar{H}^\perp$. For such a pair, from the very definition you can deduce that $\mathbb{E}[W(h)\cdot W(h')]=0$. Therefore, $W(h)$ and $W(h')$ are uncorrelated r.v.'s and since their joint distribution is Gaussian it follows their independence.