The most important metric spaces are normed vector spaces, which we want to introduce now.
Let \(V\) be a vector space over the field of real numbers \(\mathbb R\). A norm is a map. \[\|~\|: \begin{cases} V&\mapsto \mathbb R\\ x&\mapsto \|x\|\\ \end{cases}\] with the following properties:
(1) \(\|x\|=0\) if and only if \(x=0\).
(2) \(\|\lambda x\|=|\lambda|\cdot\|x\|\) for all \(\lambda\in\mathbb R\).
(3) \(\|x+y\|\le \|x\|+\|y\|\) for all \(x,y\in V\).
If such a map exists, the pair \((V,\|~\|)\) is called a normed vector space.
Definitions: 1 2 3 4 5 6 7 8
Parts: 9 10
Proofs: 11 12 13
Propositions: 14 15 16