Weyl's Unitarity Trick
 Theorem 1 (Weyl's Unitarity Trick): Let $G$ be a finite group and let $(V, \rho)$ be a representation of $G$. Then there exists an inner product on $V$ for which $(V, \rho)$ is unitary.
• Proof: Let $(V, \rho)$ be a representation of $G$. Let $\langle -, - \rangle$ be an inner product on $V$. Define a new inner product $(-, -)$ on $V$ defined for all $v, w \in V$ by:
• Since $G$ is a finite group, the above sum has no convergence issues. Moreover, $(-, -)$ is indeed an inner product since $\langle -, - \rangle$ is an inner product.
• Lastly, $(V, \rho)$ with respect to the inner product $(-, -)$ is unitary since for all $h \in G$ and for all $v, w \in V$ we have that:
• (Where the equality at $(*)$ comes from the fact that for each fixed $h \in G$, the map $g \to gh$ is an automorphism of $G$, and so the sums are just rearrangements of one another). $\blacksquare$