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While the bilinearity (or sesquilinearity) of the averaged form is immediate from the definition, the video explicitly leaves the verification of positive definiteness as an exercise. One must show that $(v, v) > 0$ for all nonzero $v \in V$ to confirm that the averaged form satisfies all inner product axioms.

Conditions: The averaged form is defined as $(v, w) := \frac{1}{|G|} \sum_{g \in G} \langle \rho(g)v, \rho(g)w \rangle$.; $\langle \cdot, \cdot \rangle$ is a valid inner product on $V$.; $G$ is finite.

While the bilinearity (or sesquilinearity) of the averaged form is immediate from the definition, the video explicitly leaves the verification of positive definiteness as an exercise. One must show that $(v, v) > 0$ for all nonzero $v \in V$ to confirm that the averaged form satisfies all inner product axioms.

Conditions: The averaged form is defined as $(v, w) := \frac{1}{|G|} \sum_{g \in G} \langle \rho(g)v, \rho(g)w \rangle$.; $\langle \cdot, \cdot \rangle$ is a valid inner product on $V$.; $G$ is finite.