is this equality true?

Question

Let $\{M_\alpha\}$ be a family of closed subspaces of a Hilbert space H. Is this equality true? ${(\cap M_\alpha^\bot)^\bot}=\overline{span(\cup M_\alpha)}$.

thanks for your help.

Answer 1

Yes. This follows from $A^{\perp\perp} = \overline{span(A)}$ for any $A\subseteq H$ and $(\bigcup_\alpha A_\alpha)^\perp = \bigcap_\alpha A_\alpha^\perp$.