Intersection of Orthocomplements is Orthocomplement of Closed Linear Span

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Theorem

Let $H$ be a Hilbert space.

Let $\family {M_i}_{i \mathop \in I}$ be an $I$-indexed family of closed linear subspaces of $H$.


Then:

$\ds \bigcap_{i \mathop \in I} M_i^\perp = \paren {\vee \set {M_i : i \in I} }^\perp$

where:

$\perp$ denotes orthocomplementation
$\vee$ denotes closed linear span.


Corollary

Furthermore, the following equality holds:

$\ds \paren {\bigcap_{i \mathop \in I} M_i}^\perp = \vee \set {M_i^\perp : i \in I}$


Proof

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

$\ds \bigcap_{i \mathop \in I} M_i^\perp \subseteq \paren {\vee \set {M_i : i \in I} }^\perp$
$\paren {\vee \set {M_i : i \in I} }^\perp \subseteq \ds \bigcap_{i \mathop \in I} M_i^\perp$


$\ds \bigcap_{i \mathop \in I} M_i^\perp$ is contained in $\paren {\vee \set {M_i : i \in I} }^\perp$

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:

$\vee \set {M_i : i \in I} \subseteq \ds \paren {\bigcap_{i \mathop \in I} M_i^\perp }^\perp$

For $h \in \ds \bigcap_{i \mathop \in I} M_i^\perp$, by definition one has:

$h \perp M_i$ for all $i \in I$

That is:

$M_i \perp \ds \bigcap_{i \mathop \in I} M_i^\perp$

This is equivalent to saying that:

$M_i \subseteq \ds \paren {\bigcap_{i \mathop \in I} M_i^\perp }^\perp$

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

$\Box$


$\paren {\vee \set {M_i : i \in I} }^\perp$ is contained in $\ds \bigcap_{i \mathop \in I} M_i^\perp$

By definition $(2)$ of closed linear span:

$\forall i \in I: M_i \subseteq \vee \set {M_i^\perp : i \in I}$

By Orthocomplement Reverses Subset, it follows that:

$\forall i \ni I: \paren {\vee \set {M_i : i \in I} }^\perp \subseteq M_i^\perp$

Therefore, by definition of set intersection:

$\paren {\vee \set {M_i : i \in I} }^\perp \subseteq \ds \bigcap_{i \mathop \in I} M_i^\perp$

$\Box$


Thus we have established that:

$h \in \paren {\vee \set {M_i : i \in I} }^\perp \iff h \in \ds \bigcap_{i \mathop \in I} M_i^\perp$

From the definition of set equality, it follows that:

$\ds \bigcap_{i \mathop \in I} M_i^\perp = \paren {\vee \set {M_i : i \in I} }^\perp$

$\blacksquare$


Sources

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