# Continuous linear representation of compact group over complex numbers has invariant Hermitian inner product

## Statement

### For finite-dimensional vector space

Suppose $G$ is a Compact group (?) and $\alpha:G \to GL(V)$ (with $V$ finite-dimensional) is a Continuous linear representation (?) of $G$ over a vector space $V$ over $\mathbb{C}$, the field of complex numbers. Then, there exists a Hermitian inner product $b: V \times V \to V$ such that

$\! b(\alpha(g)v,\alpha(g)w) = b(v,w) \ \forall \ g \in G, \ v,w \in V$

In other words, we can think of $b$ as a Hermitian inner product invariant under the action of $G$.

Further, if we choose a basis for $V$ that is an orthonormal basis for $b$, then in this basis, all the matrices for $\alpha(g), g \in G$, are unitary matrices, i.e., they live in the group $U(n,\mathbb{C})$ if $n$ is the dimension of $V$.