Sum with orthogonal complement always closed only in Hilbert spaces?

Question

Is the following statement true? If $V$ is an inner product space over $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}$ such that $U + U^{\perp}$ is a closed subspace of $V$ for every closed subspace $U \subseteq V$, then $V$ is a Hilbert space.