Norm

Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $V$ be a vector space over $\mathbb{K}$. A norm on $V$ is a map $\norm{\cdot} \colon V \to \R$ such that, for all $x, y \in V$ and $a \in \mathbb{K}$:

1. $\norm{x + y} \le \norm{x} + \norm{y}$.
2. $\norm{ax} = \abs{a} \norm{x}$.
3. If $\norm{x} = 0$, then $x = 0$.

Remarks

These conditions imply $\norm{x} \ge 0$ for all $x$. On $\R^n$, the Euclidean norm $\sqrt{x_1^2 + \cdots + x_n^2}$, the norm $\abs{x_1} + \cdots + \abs{x_n}$ and the norm $\max_i \abs{x_i}$ are norms. Each inner product gives a norm, see the induced norm. A norm defines a metric $d(x, y) = \norm{x - y}$.

Explanations

L1 human

A norm measures the length of a vector. The length of a sum is at most the sum of the lengths, and scaling a vector scales its length. Only the zero vector has length zero.