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

## 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.