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Let $(E,g)$ be a real oriented inner product space with orthonormal basis $(e_1, \dots, e_n)$ with corresponding dual basis $(e^1, \dots, e^n)$. Then, for any $\beta \in \Lambda^0(V) := \mathbb{R}$, how does one prove that $$ * \beta = \beta \omega_E $$ where $\omega_E := e^1 \wedge \cdots \wedge e^n$ denotes the induced volume form for $(V,g)$?

I welcome feedback about the correctness of my approach and any alternate approaches that might be more efficient.

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