$p$-norm and $p$-metric¶

Definition¶

Let $\boldsymbol{x}=(x_1,x_2, \ldots,x_n)\in\mathbb{R}^n$${\mathbb{R}}^n$. For $p\in[1,+\infty)$$[1,+\infty )$, the $p$-norm, denoted $\|\cdot\|_p$${\Vert }_p$, is defined on $\mathbb{R}^n$ by:

For $p=+\infty$ one sets $\|\boldsymbol{x}\|_\infty=\max\limits_{1\le i\le n}|x_i|$$\underset{1\le i\le n}{\max }|x_i|$, which is the limit of $\|\boldsymbol{x}\|_p$ as $p\to+\infty$$+\infty$.

Correspondingly, the $p$-metric $d_p$ is defined as $d_p(\boldsymbol{x},\boldsymbol{y})=\|\boldsymbol{x}-\boldsymbol{y}\|_p$$\mathbf{y}{\Vert }_p$, that is,

The formula uses Exponentiation and Power twice: raise the coordinate magnitudes to $p$, then take the $1/p$ power of their sum. The value $+\infty$ is an Extended Real Number indexing the separate maximum-norm limit, not a finite exponent to substitute directly. Setting $p=2$ recovers the Euclidean Norm.

Interpretation¶

The $p$-norm is a way of measuring the size of a vector by raising each coordinate's absolute value to the $p$-th power, summing them, and taking the $p$-th root. The exponent decides how much weight the large coordinates get: $p=1$ adds all coordinates equally, $p=2$ gives the Euclidean length, and as $p\to+\infty$ only the largest coordinate counts.

The figure compares the unit balls $\{\boldsymbol{x}\colon\|\boldsymbol{x}\|_p\le1\}$ for $p=1,2,4,\infty$$1,2,4,\infty$. As $p$ increases, these balls expand from the diamond towards the square, because $\|\boldsymbol{x}\|_q\le\|\boldsymbol{x}\|_p$ whenever $p\le q$$q$. All four boundaries pass through $(\pm1,0)$ and $(0,\pm1)$: a vector with a single nonzero coordinate has the same $p$-norm for every $p$.

The restriction $p\ge1$ is needed for the triangle inequality. For $p=\tfrac{1}{2}$ the formula gives $\|(1,0)\|_{1/2}=\|(0,1)\|_{1/2}=1$ but $\|(1,1)\|_{1/2}=(1+1)^2=4>1+1$$1$. Geometrically, the set $\{\boldsymbol{x}\colon\|\boldsymbol{x}\|_{1/2}\le1\}$ is pinched inwards between the axes: it contains $(1,0)$ and $(0,1)$, but not the midpoint $(\tfrac{1}{2},\tfrac{1}{2})$, whose value is $2$.

Different $p$ give different distances but the same analysis: from $\|\boldsymbol{x}\|_\infty\le\|\boldsymbol{x}\|_p\le n^{1/p}\|\boldsymbol{x}\|_\infty$$n^{1/p}\Vert \mathbf{x}{\Vert }_{\infty }$, all $p$-norms on $\mathbb{R}^n$ are equivalent, and the $p$-metrics are strongly equivalent, with the same convergent sequences.

A common slip is to drop the outer root and use $\displaystyle \sum_i|x_i|^p$ as the norm. For $p>1$ this is not a norm: it scales like $|c|^p$ instead of $|c|$, so absolute homogeneity fails.

Key Results¶

• The smallest permissible value for $p$ in the $p$-norm is $1$.

Examples¶

• The Euclidean distance $d_2$ is a particular $p$-metric when $p=2$$2$.
• The taxicab distance $d_1$ is a particular $p$-metric when $p=1$$1$.
• The maximum norm $\|\cdot\|_\infty$ is a particular $p$-norm when $p=+\infty$$+\infty$.

Prerequisites¶

• Exponentiation and Power
• Extended Real Number
• Euclidean Norm

Related Concepts¶

• Euclidean Distance
• Taxicab Metric
• Maximum Norm
• Strongly Equivalent Metrics
• Discrete Metric
• Norm and Normed Vector Space

Sources¶

• Math III: Analysis · Lecture 4

Aliases: p norm, p metric, Minkowski distance Symbols: element of, infinity, real numbers, sum

Last updated: 22 October 2026 19:04