Let \(V\) be a vector space over \(\mathbb{R}\) or \(\mathbb{C}\). A map \(\lVert \cdot\rVert\colon V \to \mathbb{R}\) is called norm if it satisfies for all \(v,w\in V\) and \(\alpha\in F\)

\(\lVert v\rVert>0\) if \(v\neq 0\)

1. triangle inequality, i.e. \(\lVert v+w\rVert\le \lVert v\rVert+\lVert w\rVert\),
2. homogeneity, i.e. \(\lVert \alpha v\rVert=\lvert \alpha\rvert \lVert v\rVert\),
3. positive definiteness, i.e. \(\lVert v\rVert=0\) iff \(v=0\),
4. non-negativity, i.e. \(\lvert v\rvert>0\) for \(v\neq 0\).

See also Link to heading

• normed space
• quasinorm
• matrix norm