The Statement: Viewing $\mathbb{C}^n$ as an $n$-dimensional Hilbert space, we can view $U(n)$ as the group of linear isometries of $\mathbb{C}^n$.
(Definition: The unitary group $U(n)$ consists of all $A\in GL(n, \mathbb{C})$ with $AA^*=A^*A=I$, where $A^*=(\overline{A})^t$ is the adjoint matrix of an $n\times n$ matrix $A$.)
Source: Classical Descriptive Set Theory by Alexander S. Kechris.
My Question: I do not follow the statement at all, even after figuring out what Hilbert space and linear isometry are. Any clarification will be greatly appreciated.