Show $\langle , \rangle |_W$ non degenerate $\implies$ $\langle , \rangle |_{W^\perp}$ non degenerate

Question

Let $W \subset V$ be a subspace and $\dim V < \infty$. If $\langle , \rangle$ and the restriction $\langle , \rangle |_W$ are non degenerate, then $\langle , \rangle |_{W^\perp}$ is non degenerate as well and $(W^\perp)^\perp = W$.

Are there counterexamples if one of the conditions is not met, and how can it be proven?