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:

1. 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}$$0$.
2. Absolute homogeneity (absolute scalability): For any scalar $c \in \mathbb{F}$ and any $\boldsymbol{x} \in V$$V$, $\|c\boldsymbol{x}\| = |c|\cdot\|\boldsymbol{x}\|$$\Vert \mathbf{x}\Vert$, where $|c|$ is the absolute value (modulus) of $c$.
3. Triangle inequality: For any $\boldsymbol{x}, \boldsymbol{y} \in V$$V$: $\|\boldsymbol{x} + \boldsymbol{y}\| \leq \|\boldsymbol{x}\| + \|\boldsymbol{y}\|$$\Vert \mathbf{y}\Vert$.

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

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.

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}\|$$\Vert \mathbf{y}\Vert$. The lower part shows absolute homogeneity: stretching $\boldsymbol{x}$ by the factor $3$ multiplies its length by $3$.

Absolute homogeneity also covers negative scalars, so $\|-\boldsymbol{x}\|=\|\boldsymbol{x}\|$$\Vert \mathbf{x}\Vert$: a vector and its opposite have the same length. Together the three axioms make $d(\boldsymbol{x},\boldsymbol{y})=\|\boldsymbol{x}-\boldsymbol{y}\|$ a metric, and a metric of this form has two properties that a general metric lacks: it does not change under translation, $d(\boldsymbol{x}+\boldsymbol{z},\boldsymbol{y}+\boldsymbol{z})=d(\boldsymbol{x},\boldsymbol{y})$$d(\mathbf{x},\mathbf{y})$, and it scales, $d(c\boldsymbol{x},c\boldsymbol{y})=|c|\,d(\boldsymbol{x},\boldsymbol{y})$$|c|d(\mathbf{x},\mathbf{y})$. The discrete metric does not scale, since $d_{\mathrm{disc}}(2\boldsymbol{x},2\boldsymbol{y})=1$ for $\boldsymbol{x}\ne\boldsymbol{y}$$\mathbf{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\|)$$\Vert )$, 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¶

• Vector Space
• Triangle Inequality
• Euclidean Norm

Related Concepts¶

• $p$-norm and $p$-metric
• Metric and Metric Space

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