How to find all isometries of Hilbert space?

Question

We know all isometries of $\mathbb R^n $ are composition of transfer by orthogonal linear functions.

How to find all surjective isometries of Hilbert space?

Is there similarity?

Answer 1

norm induced by inner product is strictly subadditive.

By mazur-ulam theorem every surjective isometry of hilbert space is composition of a transfer and a linear function. So every surjective isometry of hilbert spaces are determined.

Source: Functional Analysis of Peter.Lax page 47 for theorem 9 and page 61 for theorem 10