Define $\boxed{\|v\|=\sqrt{\langle v,v\rangle}}$. Positive definiteness gives nonnegativity and separation of zero, and bilinearity gives absolute homogeneity. <Cauchy sequence>[Cauchy]–Schwarz follows by minimizing the nonnegative quadratic $\|u-tv\|^2$ in real $t$. It then gives $\|u+v\|^2\le(\|u\|+\|v\|)^2$, establishing the triangle inequality. Thus this is a norm.

Solution

Past exam of the Probability, statistics and options II course of the University of Oxford / 2008 / P / 3 / a / Solution

 Solution (source code)