It is clear from Projection theorem that if $Z$ is a subspace, then since $X=Z\oplus Z^{\perp}$, $Z^{\perp}$ is not trivial (By the way, is there any reasoning that would show this without referring to Projection theorem? - I feel like using something unnecessarily advanced to show something trivial)
Is the same true for any closed subset of $Z$? I feel like the answer is no, but I can't find an example.