Real Norm

Why

We generalize our notion of size to $n$-dimensional space.

Definition

The norm (or Euclidean norm) of $x \in \R ^n$ is

\[ \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2}. \]

A vector $u \in \R ^n$ with $\norm{u} = 1$ is called a unit vector.

Notation

We denote the norm of $x$ by $\norm{x}$. In other words, $\norm{\cdot}: \R ^n \to \R $ is a function from vectors to real numbers. The notation follows the notation of absolute value, the magnitude of a real number, and the double verticals remind us that $x$ is a vector. A warning: some authors write $\abs{x}$ for the norm of $x$ when it is understood that $x \in \R ^n$.

We understand the norm of $x$ by comparison with the distance function $d: \R ^n \times \R ^n \to \R $. On one hand, the norm of $x$ is $d(x, 0)$. So $\norm{x}$ measures the length of the vector $x$ from the origin $0$. On the other hand, $d(x, y) = \norm{x - y}$. So $\norm{x - y}$ measures the distance between $x$ and $y$.

Properties

The norm has several important properties:

$\norm{\alpha x} = \abs{\alpha}\norm{x}$, called (absolute) homogeneity,
$\norm{x + y} \leq \norm{x} + \norm{y}$, called the triangle inequality,
$\norm{x} \geq 0$, called non-negativity, and
$\norm{x} = 0 \iff x = 0$, called definiteness.