$\displaystyle U \subseteq R^n$ and $\displaystyle U^\perp$ is it's orthogonal complement.
Is there a case where
$\displaystyle {(U^\perp)}^\perp \neq U$
Thanks.
Yes.
In your first case, you chose $\displaystyle U \subseteq R^n$ such that U may or may not be a subspace. By definition of orthogonal complement, $\displaystyle U^\perp$ is a subspace. So $\displaystyle {(U^\perp)}^\perp$ is also a subspace (of $\displaystyle R^n$), and we know $\displaystyle U \subseteq U^\perp$ and in fact $\displaystyle U^\perp$ is the subspace spanned by U.
For your second case, you chose to start with $\displaystyle U^\perp$ which is a subspace of $\displaystyle R^n$, so $\displaystyle {({(U^\perp)}^\perp)}^\perp$ is the subspace spanned by $\displaystyle U^\perp$, but that implies equality based on the definition of spanned.