Norm

• categories: Real Analysis, Definition

The norm is a function $||\cdot ||:V\to \mathbb{R}$ that assigns a non-negative real number to each vector in $V$, satisfying the following properties for all $u,v\in V$ and any scalar $\alpha$:

1. Non-negativity: $\mid \mid v\mid \mid =0⟺v=0$
2. Scalar multiplication: $\mid \mid \alpha v\mid \mid =\mid \alpha \mid \cdot \mid \mid v\mid \mid$
3. Triangle inequality:$\mid \mid u+v\mid \mid \le \mid \mid u\mid \mid +\mid \mid v\mid \mid$