The Hodge star on an oriented finite-dimensional real inner-product space

Let $V$ be an oriented finite-dimensional real inner product space of dimension $n$, with Gram pairings on the exterior powers of The Gram inner product on ${\mathrm{\Lambda }}^{k}V$ and oriented unit volume form $\omega$ of The oriented unit volume form. For each degree $k$ with $0\le k\le n$, the Hodge star

$\star :{\mathrm{\Lambda }}^{k}V⟶{\mathrm{\Lambda }}^{n-k}V$

is the linear map characterized by

$\alpha \wedge \star \beta =⟨\alpha ,\beta ⟩\omega \text{for all }\alpha ,\beta \in {\mathrm{\Lambda }}^{k}V\mathrm{.}$

Existence and uniqueness of $\star$ for every degree $0\le k\le n$ are proved in The Hodge star exists uniquely and is given by the complementary-basis formula in an oriented orthonormal basis ↗, which discharges the well-definedness obligation recorded above.

The Hodge star needs the metric (through the Gram pairing) and the orientation (through $\omega$); the bare vector-space structure does not determine it. With the opposite orientation $\omega$ is replaced by $-\omega$, and the star is replaced by $-\star$.