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;

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

$V = S \oplus S^\perp$

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Pseudo-Riemannian Metrics