The Induced Norm is a Norm, and Induces a Metric

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space, let $\lVert\cdot\rVert$ be the norm induced by the inner product, and write $u-v=u+(-v)$ with the additive inverse of Elementary Identities in a Vector Space. Then the following hold.

1. (Cauchy-Schwarz in norm form) For all $u,v\in V$, the inequality of Cauchy-Schwarz Inequality in a Complex Inner Product Space takes the form

with the modulus on the left.

2. (Norm) $\lVert\cdot\rVert$ is a norm on $V$; that is, it satisfies positivity, absolute homogeneity and the triangle inequality.

3. (Induced metric) The map assigning to each pair $u,v$ of elements of $V$ the real number $d(u,v)=\lVert u-v\rVert$ is a metric on $V$.