A <norm> on a real <vector space> $V$ is a map $p:V\to[0,\infty)$ satisfying $p(x)=0$ exactly when $x=0$, $p(tx)=|t|p(x)$ for real $t$, and $p(x+y)\le p(x)+p(y)$. Its induced <metric> is $d_p(x,y)=p(x-y)$. The three axioms respectively give definiteness, compatibility with scaling, and the <triangle inequality>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2001 / ib / Paper 3 / 1A / Solution

 Solution (source code)