For vector space \(V\) on field \(\mathbb{R}\) a norm is a function \(||\cdot||:V\rightarrow\mathbb{R}\) satisfying the following properties for \(v,x\in V\): 1. \(\Vert v\Vert \geq 0~\forall v\in V\) 2. \(||v||=0\) iff \(v=0\) 3. \(\Vert\alpha v \Vert =|\alpha|\Vert v\Vert\) for \(\alpha\in\mathbb{R}\) 4. \(\Vert v+z\Vert\leq \Vert v\Vert + \Vert x \Vert\) (triangle inequality)