Unitary Operator

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space and let $T$ be a linear operator on $V$.

The operator $T$ is unitary if it is surjective, that is, every $v\in V$ satisfies $v=T(u)$ for some $u\in V$, and if

$$\langle T(u),T(v)\rangle=\langle u,v\rangle\qquad\text{for all }u,v\in V.$$