Properties of Unitary Operators

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space, with induced norm $\lVert\cdot\rVert$, which is a norm on $V$ by claim 2 of The Induced Norm is a Norm, and Induces a Metric. Let $T$ be a unitary operator on $V$. Throughout, $\lVert\cdot\rVert$ without a subscript denotes the norm of a vector of $V$ and $\lVert\cdot\rVert_{\mathrm{op}}$ the operator norm of a bounded linear operator on $V$; the order is that of the ordered field of real numbers. Then the following hold.

1. (Preservation of the norm) For every $u\in V$,

$$\lVert T(u)\rVert=\lVert u\rVert .$$

2. (Bijectivity) $T$ is a bijection from $V$ onto $V$.

3. (Boundedness and operator norm) $T$ is a bounded linear operator on $V$; consequently it has exactly one operator norm $\lVert T\rVert_{\mathrm{op}}$ by Existence and Uniqueness of the Operator Norm, and

$$\lVert T\rVert_{\mathrm{op}}\le1 .$$