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:


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

Proof
By definition of set equality, it suffices to prove the following two inclusions:


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

$\displaystyle \bigcap_{i \in I} M_i^\perp$ is contained in $\left({\vee \left\{{M_i : i \in I}\right\}}\right)^\perp$
By Orthocomplement is Closed Linear Subspace and Closed Linear Subspaces Closed under Intersection, both spaces considered are closed linear subspaces of $H$.

By Orthocomplement Reverses Subset, the required containment is equivalent to:


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

For $h \in \displaystyle \bigcap_{i \in I} M_i^\perp$, by definition one has $h \perp M_i$ for all $i \in I$.

That is, $M_i \perp \displaystyle \bigcap_{i \in I} M_i^\perp$; this is equivalent to saying that $M_i \subseteq \displaystyle \left({\bigcap_{i \in I} M_i^\perp }\right)^\perp$.

Definition $(2)$ of closed linear span now grants the desired subset relation.

$\left({\vee \left\{{M_i : i \in I}\right\}}\right)^\perp$ is contained in $\displaystyle \bigcap_{i \in I} M_i^\perp$
By definition $(2)$ of closed linear span, $M_i \subseteq \vee \left\{{M_i^\perp : i \in I}\right\}$ for all $i \in I$.

By Orthocomplement Reverses Subset, it follows that, for all $i \in I$:


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

Therefore, by definition of set intersection:


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

Therefore, we have established that $h \in \left({\vee \left\{{M_i : i \in I}\right\}}\right)^\perp \iff h \in \displaystyle \bigcap_{i \in I} M_i^\perp$.

From the definition of set equality, it follows that $\displaystyle \bigcap_{i \in I} M_i^\perp = \left({\vee \left\{{M_i : i \in I}\right\}}\right)^\perp$.

Proof of Corollary
From Orthocomplement is Closed Linear Subspace, the $M_i^\perp$ are an $I$-indexed family of closed linear subspaces of $H$.

The main result gives:


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

Taking the orthocomplement of both sides, and using Corollary to Double Orthocomplement is Closed Linear Span yields the result.