How to define an inner product on $\Lambda^p\mathrm{T}^* M$ for a Riemannian manifold $(M^n,g)$

Question

In the book Hamilton's Ricci Flow, we are given the following definition for an inner product on $\Lambda^p\mathrm{T}^* M$. If $\gamma,\eta\in\Lambda^p\mathrm{T}^* M$, then we define $$\langle\gamma,\eta\rangle = p!g^{i_1 j_1}\ldots g^{i_p j_p} \gamma_{i_1\ldots i_p}\eta_{j_1\ldots j_p}.$$ But why is this positive definite? I.e., why is it true that $$\langle\gamma,\gamma\rangle = p! g^{i_1 j_1}\ldots g^{i_p j_p} \gamma_{i_1\ldots i_p}\gamma_{j_1\ldots j_p}>0$$ for $\gamma\neq 0$?

Answer 1

Because the product is coordinate invariant, so you can just take a local normal coordinate system around a point, in which the metric is diagonalized. Then the expression in local coordinates is manifestly the sums of squares of real numbers, and hence non-negative.

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Alternatively and equivalently, if you take an orthonormal frame then the inverse metric $(g^{-1}) = \sum_{i = 1}^n (e_i)\otimes (e_i)$. Then $$ \langle \gamma, \gamma\rangle = p! \sum_{i_{1} \ldots i_{p} = 1}^n \left| \gamma_{i_1 i_2 i_3 \cdots i_p}\right|^2 \geq 0$$