TITLE: How can I show $U^{\bot \bot}\subseteq \overline{U}$?
QUESTION [5 upvotes]: Let $H$ be a Hilbert space and $U$ a subspace. Let $U^{\bot}$ denote its orthogonal complement. 
I had no trouble showing $\overline{U}\subseteq U^{\bot\bot}$. But now I'm stuck for $\supseteq$.

Please could someone help me finish this argument?

This is what I am trying to do:
Let $x \in U^{\bot \bot}$. The goal is to construct a sequence $u_n\in U$ such that $u_n \to x$. 
Since $\overline{U} \subset U^{\bot \bot}$ either $x\in \overline{U}$ or $x \notin \overline{U}$. If $x \in \overline{U}$ then we're done. If $x \notin \overline{U}$ I don't know what I can do.

REPLY [2 votes]: For closed subspaces:
$$Z=\overline{Z}\implies\mathcal{H}=Z\oplus_\perp Z^\perp$$
For orthogonal decompositions:
$$\mathcal{H}=Z\oplus_\perp Z'\implies Z'=Z^\perp$$
Taking both together one gets:
$$\overline{U}=\left(\overline{U}^\perp\right)^\perp=U^{\perp\perp}$$
(This works even for the closure on the span of plain sets.)