Continuous linear representation of compact group over reals has invariant symmetric positive-definite bilinear form

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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 linear representation of G over a vector space V over \R, the field of real numbers. Then, there exists a symmetric positive-definite bilinear form 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 dot 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 orthogonal matrices. Thus, another formulation is that every continuous finite-dimensional linear representation over the real numbers is an orthogonalizable linear representation.

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