Norm — Definition, Formula & Examples

A norm is a function that assigns a non-negative length or size to a vector in a vector space. The most common norm is the Euclidean norm, which gives the straight-line distance from the origin to the tip of the vector.

A norm on a vector space $V$ over $\mathbb{R}$ (or $\mathbb{C}$ ) is a function $\|\cdot\|: V \to \mathbb{R}$ satisfying three axioms: (1) $\|\mathbf{v}\| \geq 0$ with equality if and only if $\mathbf{v} = \mathbf{0}$ (positive definiteness), (2) $\|c\mathbf{v}\| = |c|\,\|\mathbf{v}\|$ for any scalar $c$ (absolute homogeneity), and (3) $\|\mathbf{u} + \mathbf{v}\| \leq \|\mathbf{u}\| + \|\mathbf{v}\|$ (triangle inequality).

Key Formula

\|\mathbf{v}\|_2 = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}

Where:

$\mathbf{v}$ = A vector in $\mathbb{R}^n$
$v_1, v_2, \ldots, v_n$ = The components of the vector
$\|\mathbf{v}\|_2$ = The Euclidean (L²) norm of the vector

How It Works

To compute the Euclidean norm (also called the

L^2

norm or 2-norm), square each component of the vector, sum the squares, and take the square root. Other norms exist: the

L^1

norm sums the absolute values of the components, and the

L^\infty

norm takes the largest absolute component value. All norms satisfy the same three axioms, but they measure "size" differently. Which norm you use depends on the application — the Euclidean norm captures geometric distance, while the

L^1

norm is common in optimization and data science.

Worked Example

Problem: Find the Euclidean norm of the vector

\mathbf{v} = (3, -4, 12)

Square each component: Compute the square of each entry.

3^2 = 9, \quad (-4)^2 = 16, \quad 12^2 = 144

Sum the squares: Add the results together.

9 + 16 + 144 = 169

Take the square root: Apply the square root to get the norm.

\|\mathbf{v}\|_2 = \sqrt{169} = 13

Answer:

\|\mathbf{v}\|_2 = 13

Why It Matters

Norms appear throughout applied mathematics and engineering — computing distances between data points in machine learning, measuring error in numerical methods, and defining convergence in functional analysis. Normalizing a vector (dividing by its norm to get a unit vector) is a routine operation in physics, computer graphics, and signal processing.

Common Mistakes

Mistake: Forgetting to square the components before summing, and instead summing the absolute values.

Correction: That computes the

L^1

norm, not the Euclidean norm. For the

L^2

norm, you must square each component, sum, then take the square root.