Norm

linear-algebra

Definition

Norm

Let $V$ be a vector space over a scalar field $F$. A norm is a function $||\cdot ||:V\to \mathbb{R}$ that assigns a real number to every vector with the following properties for all vectors $x,y\in V$ and scalar $\alpha \in F$:

• Positive Definiteness:

$$\begin{array}{rl}||x||& \ge 0\\ ||x||& =0⟺x=0\end{array}$$

• Absolute Homogeneity:

$$||\alpha x||=|\alpha |\cdot ||x||$$

• Triangle Inequality:

$$||x+y++\le ||x||+||y||$$