Norm Induced by a Complex Inner Product

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space and let $v\in V$. By condition 4 of Complex Inner Product Space the number $\langle v,v\rangle$ is a real number with $0\le\langle v,v\rangle$.

The norm induced by the inner product assigns to $v$ the unique real number $\lVert v\rVert$ with $0\le\lVert v\rVert$ and $\lVert v\rVert^{2}=\langle v,v\rangle$; such a number exists and is unique by Existence and Uniqueness of the Nonnegative Square Root.