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

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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.

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