Norm and Normed Vector Space¶

Definition¶

A norm on a vector space $V$ over the scalar field $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$ (in this course $\mathbb{R}$ ) is a function $\|\cdot\|\colon V\to[0,+\infty)$ that adheres to the following criteria:

Nonnegativity and definiteness: For any $\boldsymbol{x} \in V$ $V$ , $\|\boldsymbol{x}\| \geq 0$ $0$ , and $\|\boldsymbol{x}\| = 0$ if and only if $\boldsymbol{x} = \boldsymbol{0}$ $0$ .
Absolute homogeneity (absolute scalability): For any scalar $c \in \mathbb{F}$ and any $\boldsymbol{x} \in V$ $V$ , $\|c\boldsymbol{x}\| = |c|\cdot\|\boldsymbol{x}\|$ $∥ x ∥$ , where $|c|$ is the absolute value (modulus) of $c$ .
Triangle inequality: For any $\boldsymbol{x}, \boldsymbol{y} \in V$ $V$ : $\|\boldsymbol{x} + \boldsymbol{y}\| \leq \|\boldsymbol{x}\| + \|\boldsymbol{y}\|$ $∥ y ∥$ .

A normed vector space $(V, \|\cdot\|)$ is a vector space $V$ over $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$ , equipped with a norm $\|\cdot\|$ $∥$ .

The norm version of the Triangle Inequality bounds the length of a sum by the sum of the lengths. The Euclidean Norm $\|\boldsymbol{x}\|_2=\sqrt{x_1^2+\cdots+x_n^2}$ is the standard example satisfying all three axioms.

Interpretation¶

A norm is a rule that assigns a length to every vector in a way that fits the vector-space operations. A normed vector space is a vector space in which every vector has such a length, and scaling or adding vectors changes lengths as one would expect.

Vectors x, y and x + y forming a parallelogram, illustrating the triangle inequality, and a vector x stretched to 3x, illustrating that the norm scales with the factor.

The upper part of the figure shows the triangle inequality: $\boldsymbol{x}+\boldsymbol{y}$ is the diagonal of the parallelogram spanned by $\boldsymbol{x}$ and $\boldsymbol{y}$ , and its length is at most $\|\boldsymbol{x}\|+\|\boldsymbol{y}\|$ $∥ y ∥$ . The lower part shows absolute homogeneity: stretching $\boldsymbol{x}$ by the factor $3$ multiplies its length by $3$ .

Each $p\ge1$ gives a norm on $\mathbb R^2$ with its own unit circle; all of them are convex and symmetric about $\boldsymbol 0$ .

Absolute homogeneity also covers negative scalars, so $\|-\boldsymbol{x}\|=\|\boldsymbol{x}\|$ $∥ x ∥$ : a vector and its opposite have the same length. Together the three axioms make $d(\boldsymbol{x},\boldsymbol{y})=\|\boldsymbol{x}-\boldsymbol{y}\|$ a metric, and a metric of this form has two properties that a general metric lacks: it does not change under translation, $d(\boldsymbol{x}+\boldsymbol{z},\boldsymbol{y}+\boldsymbol{z})=d(\boldsymbol{x},\boldsymbol{y})$ $d (x, y)$ , and it scales, $d(c\boldsymbol{x},c\boldsymbol{y})=|c|\,d(\boldsymbol{x},\boldsymbol{y})$ $∣ c ∣ d (x, y)$ . The discrete metric does not scale, since $d_{\mathrm{disc}}(2\boldsymbol{x},2\boldsymbol{y})=1$ for $\boldsymbol{x}\ne\boldsymbol{y}$ $y$ , so it comes from no norm.

The Euclidean norm from Math II is one norm among many. The $p$ -norms, including the taxicab norm and the maximum norm, are others on $\mathbb{R}^n$ , and the uniform norm measures bounded functions.

Norms return in Math V, where the largest singular value of a matrix serves as a length for matrices.

Key Results¶

Given a normed vector space $(V, \|\cdot\|)$ $∥)$ , one can define the metric as $d(\boldsymbol{x},\boldsymbol{y}) = \|\boldsymbol{x} - \boldsymbol{y}\|,$ making $(V, d)$ inherently a metric space.

Examples¶

The real line $\mathbb{R}$ with the absolute-value norm forms a normed vector space over $\mathbb{R}$ .
The plane $\mathbb{R}^2$ with the Euclidean norm is similarly a normed vector space. Using the taxicab norm, $\mathbb{R}^2$ remains a normed vector space.

Prerequisites¶

Sources¶

Math III: Analysis · Lecture 4

Aliases: normed space, normed vector space Keywords: norm Symbols: complex numbers, element of, infinity, real numbers, square root

Last updated: 20 October 2026 03:02