Inner product on exterior powers of a cotangent space ($⟨{\xi }_1\wedge \cdots \wedge {\xi }_p,{\zeta }_1\wedge \cdots \wedge {\zeta }_p⟩=\det (⟨{\xi }_i,{\zeta }_j⟩)$)

A Riemannian metric induces a dual inner product on each cotangent space. On decomposable covectors define $⟨{\xi }_1\wedge \cdots \wedge {\xi }_p,{\zeta }_1\wedge \cdots \wedge {\zeta }_p⟩=\det (⟨{\xi }_i,{\zeta }_j⟩)$ and extend bilinearly. Increasing wedges of an orthonormal coframe form an orthonormal basis. This positive-definite inner product is the one used to define the Hodge star operator.