Equivalent Properties of Nondegenerate Subspace of Scalar Product Space

Theorem
Let $\struct {V, q}$ be a scalar product space.

Let $S \subseteq V$ be a linear subspace.

Let $S^\perp$ be the vector subspace perpendicular to $S$ with respect to $q$.

Then the following are equivalent:


 * $S$ is nondegenerate;


 * $S^\perp$ is nondegenerate;


 * $S \cap S^\perp = \set 0$


 * $V = S \oplus S^\perp$