Intersection of Orthocomplements is Orthocomplement of Closed Linear Span

Theorem
Let $H$ be a Hilbert space.

Let $\left({M_i}\right)_{i \in I}$ be an $I$-indexed set of closed linear subspaces of $H$.

Then:


 * $\displaystyle \bigcap_{i \in I} M_i^\perp = \left({\vee \left\{{M_i : i \in I}\right\}}\right)^\perp$

where $\perp$ denotes orthocomplementation, and $\vee$ denotes closed linear span.

Corollary
Furthermore, the following equality holds:


 * $\displaystyle \left({\bigcap_{i \in I} M_i }\right)^\perp = \vee \left\{{M_i^\perp : i \in I}\right\}$