# Definition A *normed vector space* $V$ is a [[Vector Space]] with a [[Map|mapping]] $n:V\rightarrow\mathbb{R}$ that satisfies the axioms: 1. $n(\pmb{x}) \geq 0 \forall \pmb{x} \in V \quad \text{and} \quad n(\pmb{x}) = 0 \iff \pmb{x} = 0$ 2. $n(a\pmb{x}) = |a|n(\pmb{x}) \quad \forall a \in \mathbb{R} \quad \forall \pmb{x} \in V$ 3. $n(\pmb{x} + \pmb{y}) \leq n(\pmb{x}) + n(\pmb{y}) \quad \forall \pmb{x}, \pmb{y} \in V$ This mapping is called the norm.