Remark

Let E be a Hausdorff prehilbertian space, $(V_i)_{i \in I}$ a family of vector subspaces of E such that for every pair of distinct indices $i, k$, the subspaces $V_i$ and $V_k$ are orthogonal. Then, for every $k \in I$, the intersection of $V_k$ and of the closed vector subspace $W_k$ generated by the union of the $V_i$ for all $i \neq k$ reduces to 0 for, if $x$ belongs to $V_k$ and also to $W_k$, then it is orthogonal to all the $V_i$ for $i \neq k$, hence to $W_k$. In particular, $x$ is orthogonal to itself, hence is zero.