Intersection of Orthocomplements is Orthocomplement of Closed Linear Span/Corollary

Corollary to Intersection of Orthocomplements is Orthocomplement of Closed Linear Span

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:

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

where:

$\perp$ denotes orthocomplementation

$\vee$ denotes closed linear span.

Proof

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

From Intersection of Orthocomplements is Orthocomplement of Closed Linear Span:

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

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

$\blacksquare$

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{II}.3$ Exercise $3$