Norm

Definition

A norm on a vector space $V$ is a function $\|\cdot\|: V \to \mathbb{R}$ satisfying:

1. Positive definiteness: $\|x\| \geq 0$, with equality iff $x = 0$
2. Absolute homogeneity: $\|\alpha x\| = |\alpha| \|x\|$ for all scalars $\alpha$
3. Triangle inequality: $\|x + y\| \leq \|x\| + \|y\|$

From Inner Products

Given an inner product, the induced norm is:

$$ \|x\| = \sqrt{\langle x, x \rangle} $$

For functions:

$$ \|f\| = \sqrt{\int |f(x)|^2 \, dx} $$

Intuition

The norm measures the “size” or “magnitude” of a vector or function:

• In physics, $\|f\|^2$ often represents energy
• The norm generalizes the concept of length from Euclidean space

Common Norms

Norm	Definition	Name
$\|x\|_1 = \sum \lvert x_i \rvert$	Sum of absolute values	$L^1$ / Manhattan
$\|x\|_2 = \sqrt{\sum \lvert x_i \rvert^2}$	Square root of sum of squares	$L^2$ / Euclidean
$\|x\|_\infty = \max \lvert x_i \rvert$	Maximum absolute value	$L^\infty$ / Supremum

Related Concepts

• inner_product — induces a norm via $\|x\| = \sqrt{\langle x, x \rangle}$
• Lp_space — function spaces defined by $L^p$ norms
• Metric — a norm induces a metric via $d(x,y) = \|x - y\|$