Let \((V,\|~\|)\) be a normed vector space. Then \[d(x,y):=\|x-y\|,~~~~~~~(x,y\in V)\] defines a distance on \(V\)^{1}.
Proofs: 1
We also may use the notation \(\|x,y\|\) instead of \(\|x-y\|\), since \(\|x-y\|=\|y-x\|\) and it does not matter if we express the distance as \(\|x-y\|\) or \(\|y-x\|\). ↩