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

From Groupprops

## Statement

### For finite-dimensional vector space

Suppose is a Compact group (?) and (with finite-dimensional) is a Continuous linear representation (?) of over a vector space over , the field of complex numbers. Then, there exists a Hermitian inner product such that

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

Further, if we choose a basis for that is an orthonormal basis for , then in this basis, all the matrices for , are unitary matrices, i.e., they live in the group if is the dimension of .

## Related facts

- Continuous linear representation of compact group over reals has invariant symmetric positive-definite bilinear form
- Linear representation of finite group over reals has invariant symmetric positive-definite bilinear form
- Linear representation of finite group over complex numbers has invariant Hermitian inner product