Hodge star is a smooth bundle isomorphism

For $n>0$, take a local positive smooth coframe and apply Gram–Schmidt: subtract its projections on preceding normalized covectors and divide by the positive norm of the residual. Independence ensures nonzero residuals, and positive square roots make the coefficients smooth. This gives an oriented orthonormal coframe $e^1,\dots ,e^n$. For each increasing multi-index $I$, define $\ast e^I=ϵ(I,I^c)e^{I^c}$, where $e^I\wedge e^{I^c}=ϵ(I,I^c)e^1\wedge \cdots \wedge e^n$.

For an increasing $J$ of size $k$, $e^J\wedge \ast e^I$ is zero unless $J=I$, because otherwise an index repeats, and when $J=I$ it equals the positive volume form. Thus it equals $⟨e^J,e^I⟩{\mathrm{vol}}_g$. Bilinearity proves the defining equation for arbitrary covectors. If two proposed images of $\beta$ differ, expansion of their difference in the complementary wedge basis and pairing with each $e^I$ forces every coefficient zero. Hence the image is unique.

Uniqueness makes local formulas agree on overlaps. Their matrices in these smooth frames are signed permutations, hence smooth and invertible, with smooth inverse matrices. For $n=0$, the map on the scalar fibre is $a\mapsto \epsilon a$ with locally constant orientation sign $\epsilon$; it satisfies the equation and is its own inverse. Empty base gives the empty bundle map.