Orthogonal complement turns sums into intersections.Suggest a slogan
Let \(V\) be an inner product space and \((W_i)_{i\in I}\) a family of subspaces of \(V\), where \(W^\perp\coloneqq\theset{v\in V\suchthat \inner{w}{v}=0\ \forall w\in W}\). Then \[ \qty{ \sum_{i\in I} W_i}^\perp = \bigcap_{i\in I} W_i^\perp. \]
Each \(W_i\subseteq\sum_i W_i\), so \(\qty{\sum_i W_i}^\perp\subseteq W_i^\perp\) for every \(i\). Conversely, if \(v\in\bigcap_i W_i^\perp\) and \(w=w_{i_1}+\cdots+w_{i_r}\) with \(w_{i_j}\in W_{i_j}\), then \(\inner{w}{v}=\sum_j\inner{w_{i_j}}{v}=0\).